The short answer is that, no, MusicBee will not let you tag or retag files with unsupported container tag types like that.
I'm not sure what would, except maybe something like ffmpeg
if you forced it with the right command line parameters.
The longer answer is that if you're writing your own music player, I don't think you'll find much joy in trying to support non-standard files like AAC streams with ID3 tags.
Could you use something like TagLib to handle tag reading for your player so you don't have to work out the atom metadata stuff yourself?taglib.org
Another option would be to repackage your existing M4A files into Matroska containers as mka files.
This has the dual benefit of easily readable metadata but without the quality loss you'd get from converting them to mp3s.
(note that MusicBee doesn't appear to support mka files)
Bee excellent to each other...