I had the same issue where an iTunes import/copy of an M4P file did not play. I exported from the QuickTime Player, using the "iPad, iPhone 4 & Apple TV" setting, and it successfully created a file that would play through Apple TV. (The estimated file size was way off however, actual export was much smaller and similar in size to the source, and the higher quality export option, "iPad2, iPhone 5 & Apple TV" was grayed out and not available.)
There's a good thread on this here: http://forums.macrumors.com/showthread.php?p=20884010
They talk about the issue being a 5.1 surround thing that iTunes does not handle. I'm not sure what the conversion did to the audio in my case, but I assume the conversion was lossy. See below for file info, original and conversions:
Major difference is that the original and iTunes converted files were .mp4, but the QT Player output a .m4v file that played in iTunes and my AppleTV.
All files (using get info) said they were: 720 × 404; AAC, H.264; 2 audio channels and 42:44 duration. Both MP4s (original and itunes import) were exactly the same byte count so it looks like iTunes copied and did not recompress the file, however the QT Player exported M4V file was slightly smaller in size, by about 100KB (for a 425MB file).
I don't know where/how to view the deeper audio format information to find if any of the files had surround encoding.
MacOS X 10.9.5,
AppleTV V 3,
QT V. 10.3,