I have solved pretty much the exact same problem using this SQL script. You need to modify it to fit your problem and accept there is a high amount of risk involved.

BEGIN TRANSACTION [ConvertTransaction]
BEGIN TRY
  DECLARE @FromPropertyID1 int
  DECLARE @ToPropertyID1 int
  DECLARE @FromPropertyID2 int
  DECLARE @ToPropertyID2 int
  DECLARE @FromPropertyID3 int
  DECLARE @ToPropertyID3 int
  
  DECLARE @RC int
  DECLARE @PageID int
  DECLARE @FromPageType int
  DECLARE @ToPageType int
  DECLARE @Recursive bit
  DECLARE @IsTest bit

  SELECT @PageID = 1
  SELECT @FromPageType = pkID FROM tblContentType WHERE ContentTypeGUID = '0a89e464-56d4-449f-aea8-2bf774ab8730'
  SELECT @ToPageType = pkID FROM tblContentType WHERE ContentTypeGUID = '40ed0421-2d37-4751-97dc-c7ea7033c3a5'
  
  SELECT @FromPropertyID1 = pkID FROM tblPropertyDefinition WHERE fkContentTypeID = @FromPageType AND [Name] = 'Copyright'
  SELECT @FromPropertyID2 = pkID FROM tblPropertyDefinition WHERE fkContentTypeID = @FromPageType AND [Name] = 'Description'
  SELECT @FromPropertyID3 = pkID FROM tblPropertyDefinition WHERE fkContentTypeID = @FromPageType AND [Name] = 'MediumThumbnail'
  
  SELECT @ToPropertyID1 = pkID FROM tblPropertyDefinition WHERE fkContentTypeID = @ToPageType AND [Name] = 'Copyright'
  SELECT @ToPropertyID2 = pkID FROM tblPropertyDefinition WHERE fkContentTypeID = @ToPageType AND [Name] = 'Description'
  SELECT @ToPropertyID3 = pkID FROM tblPropertyDefinition WHERE fkContentTypeID = @ToPageType AND [Name] = 'MediumThumbnail'
  
  SET @Recursive = 1
  SET @IsTest = 0
  
  print @PageID
  print @FromPageType
  print @ToPageType
  
  DECLARE @MasterLanguageID int

  SET @MasterLanguageID = 8
  
  EXECUTE @RC = [dbo].[netConvertPropertyForPageType]
     @PageID
    ,@FromPageType
    ,@FromPropertyID1
    ,@ToPropertyID1
    ,@Recursive
    ,@MasterLanguageID
    ,@IsTest

  print @RC
  
  EXECUTE @RC = [dbo].[netConvertPropertyForPageType]
     @PageID
    ,@FromPageType
    ,@FromPropertyID2
    ,@ToPropertyID2
    ,@Recursive
    ,@MasterLanguageID
    ,@IsTest

  print @RC

  EXECUTE @RC = [dbo].[netConvertPropertyForPageType]
     @PageID
    ,@FromPageType
    ,@FromPropertyID3
    ,@ToPropertyID3
    ,@Recursive
    ,@MasterLanguageID
    ,@IsTest

  print @RC
  
  EXECUTE @RC = [dbo].[netConvertPageType]
     @PageID
    ,@FromPageType
    ,@ToPageType
    ,@Recursive
    ,@IsTest
  
  print @RC

  COMMIT TRANSACTION [ConvertTransaction]
END TRY
BEGIN CATCH
  ROLLBACK TRANSACTION [ConvertTransaction]
END CATCH

-- Run separately
DECLARE @FromPageType int
SELECT @FromPageType = pkID FROM tblContentType WHERE ContentTypeGUID = '0a89e464-56d4-449f-aea8-2bf774ab8730'
EXECUTE netContentTypeDelete @ContentTypeID = @FromPageType

Has anyone had luck doing this using code only? 

I may find myself going the DB route but I am hoping to avoid that if possible.

I think you can workaround by skipping the validation - it's probably not worth doing in this case. You can use 

SaveAction.Publish | SaveAction.SkipValidation 

I had to do somehing similar for PDFS, my scheduled jon did utilise SQL as Johan suggested, the code is below and believe I found it somewhere in these forums 

    [ScheduledPlugIn(DisplayName = "Convert PDF Files", GUID = "d6619008-3e76-4886-b3c7-9a025a0c2603")]
    public class FixPdfMediaType : ScheduledJobBase
    {
        private readonly IContentRepository ContentRepository;
        private readonly IContentTypeRepository ContentTypeRepository;
        private readonly ContentMediaResolver MediaDataResolver;

        public FixPdfMediaType()
        {
            ContentRepository = ServiceLocator.Current.GetInstance<IContentRepository>();
            ContentTypeRepository = ServiceLocator.Current.GetInstance<IContentTypeRepository>();
            MediaDataResolver = ServiceLocator.Current.GetInstance<ContentMediaResolver>();
        }

        private List<BasicContent> GetPdfs()
        {
            var pdfs = new List<BasicContent>();
            var assets = ContentRepository.GetDescendents(SiteDefinition.Current.GlobalAssetsRoot);

            OnStatusChanged("Found " + assets.Count() + " assets");

            foreach (var asset in assets)
            {
                BasicContent pdf = null;
                if (ContentRepository.TryGet(asset, out pdf) && pdf.Name.EndsWith(".pdf"))
                {
                    pdfs.Add(pdf);
                }
            }

            return pdfs;
        }

        public override string Execute()
        {
            var pdfs = GetPdfs();

            OnStatusChanged("Handling " + pdfs.Count() + " PDF files");

            var output = new StringBuilder();
            int i = 0;

            //Get a suitable MediaData type from extension
            var mediaType = MediaDataResolver.GetFirstMatching(".pdf");
            var contentType = ContentTypeRepository.Load(mediaType);

            foreach (var pdf in pdfs)
            {
                if (pdf.ContentTypeID != contentType.ID)
                {
                    OnStatusChanged("Changing type for " + pdf.Name);

                    try
                    {
                        using (SqlConnection conn =
                                    new SqlConnection(ConfigurationManager.ConnectionStrings["EPiServerDB"].ConnectionString))
                        {
                            conn.Open();
                            using (SqlCommand cmd =
                                new SqlCommand("UPDATE tblPage SET fkPageTypeID=@ContentTypeID WHERE PkId = @ID", conn))
                            {
                                cmd.Parameters.AddWithValue("@ContentTypeID", contentType.ID);
                                cmd.Parameters.AddWithValue("@ID", pdf.ContentLink.ID);

                                cmd.ExecuteNonQuery();
                                i++;
                            }
                        }
                    }
                    catch (Exception ex)
                    {
                        output.AppendLine(string.Format("ID: {0}, Error: {1}",
                        pdf.ContentLink.ID,
                        ex.Message));
                    }
                }
            }

            return string.Format(
                "Fixed {0} pdfs. \n Errors:\n{1}",
                i,
                output);
        }
    }
}

For the pages admin tool DefaultPageTypeConverter is used for coverting pages which is just IContent so very similar. If you look under the covers the following happens

private DataTable ConvertPageTypeProperties(
      int pageLinkId,
      int fromPageTypeId,
      List<KeyValuePair<int, int>> propertyTypeMap,
      bool recursive,
      bool isTest)
    {
      DataTable dataTable = new DataTable("Properties");
      dataTable.Locale = CultureInfo.InvariantCulture;
      dataTable.Columns.Add("FromPropertyID");
      dataTable.Columns.Add("ToPropertyID");
      dataTable.Columns.Add("Count");
      int id = this._languageBranchRepository.Load(this._contentRepository.Get<PageData>(new ContentReference(pageLinkId)).MasterLanguage).ID;
      foreach (KeyValuePair<int, int> propertyType in propertyTypeMap)
      {
        DbCommand command = this.CreateCommand("netConvertPropertyForPageType");
        command.Parameters.Add((object) this.CreateReturnParameter());
        command.Parameters.Add((object) this.CreateParameter("PageID", (object) pageLinkId));
        command.Parameters.Add((object) this.CreateParameter("FromPageType", (object) fromPageTypeId));
        command.Parameters.Add((object) this.CreateParameter("FromPropertyID", (object) propertyType.Key));
        command.Parameters.Add((object) this.CreateParameter("ToPropertyID", (object) propertyType.Value));
        command.Parameters.Add((object) this.CreateParameter("Recursive", (object) recursive));
        command.Parameters.Add((object) this.CreateParameter("MasterLanguageID", (object) id));
        command.Parameters.Add((object) this.CreateParameter("IsTest", (object) isTest));
        command.ExecuteNonQuery();
        DataRow row = dataTable.NewRow();
        row[0] = (object) propertyType.Key;
        row[1] = (object) propertyType.Value;
        row[2] = (object) this.GetReturnValue(command);
        dataTable.Rows.Add(row);
        if (this._propertyDefinitionRepository.Load(propertyType.Key).Type.DataType == PropertyDataType.Category)
        {
          command.CommandText = "netConvertCategoryPropertyForPageType";
          command.ExecuteNonQuery();
        }
      }
      return dataTable;
    }

    private DataTable ConvertPageType(
      int pageLinkId,
      int fromPageTypeId,
      int toPageTypeId,
      bool recursive,
      bool isTest)
    {
      DataTable dataTable = new DataTable("Pages");
      dataTable.Locale = CultureInfo.InvariantCulture;
      dataTable.Columns.Add("Count");
      DbCommand command = this.CreateCommand("netConvertPageType");
      command.Parameters.Add((object) this.CreateReturnParameter());
      command.Parameters.Add((object) this.CreateParameter("PageID", (object) pageLinkId));
      command.Parameters.Add((object) this.CreateParameter("FromPageType", (object) fromPageTypeId));
      command.Parameters.Add((object) this.CreateParameter("ToPageType", (object) toPageTypeId));
      command.Parameters.Add((object) this.CreateParameter("Recursive", (object) recursive));
      command.Parameters.Add((object) this.CreateParameter("IsTest", (object) isTest));
      command.ExecuteNonQuery();
      DataRow row = dataTable.NewRow();
      row["Count"] = (object) this.GetReturnValue(command);
      dataTable.Rows.Add(row);
      return dataTable;
    }

This suggest using the database is the correct approach. I would suggest seeing if you can alter this code/stored procedures to create a more generic converter helper. But using SQL such as how Johan has suggested would likely be the best approach.

Testing out Johan's SQL I found the first part makes the conversion successfully but the second part fails with the following: 

The DELETE statement conflicted with the REFERENCE constraint "FK_tblWorkContentProperty_tblPropertyDefinition". The conflict occurred in table "dbo.tblWorkContentProperty", column 'fkPropertyDefinitionID'.

I am digging into it further but I am wondering if Johan, or anyone else who has tried something similar, ran into this issue.