You could give iVI a try ( http://www.iphonelife.com/blog/87/review-video-converter-ivi-pro-ipads-iphones ) for MTS conversion. Much cleaner, much faster than any recompression. (Note that 1080p60 videos are generally of level 4.2; with them, you'll additionally need a level decreasing.)
Thanks to Meneisys - I'll now try iVI to see if this converts the AVHCD, via a MacPro or similar. I still don't have a suitable program to use on a PC and Windows 7. Freemaker with Windows 7 does the conversion to MP4 but even though these files upload to iTunes on the PC, they're still not right. The 'convert to iPad format' on iTunes refuses to do anything saying that the MP4 file is 'already in the correct format'. Still complicated ...
You could check out the H.264 level with MediaInfo. Here's the Windows version:
Just drag the file in it and look for the number I've annotated with a red rectangle in the following screenshot:
What does it say?
iPad 3 supports file formats specially designed for iPad 3. In some cases, it also supports the standard file formats like MP4, H.264, ALAC, MP3.
There is a Total Video Converter enabling to convert video formats to iPad 3 compatible files formats to play videos in various formats on iPad 3. Bigasoft Toatl Video Converter can convert MP4 videos ad more to iPad 3 MPEG 4 Video specially for iPad 3.