So, it's been a couple of days, and I've managed to make some progress on my own. This is very, VERY far from a perfect solution, so I'd still really appreciate a more comprehensive recommendation.
First of all: After a LONG period of some quite intensive Googling, I finally found a working download link to the first of the two Mac ports of EasyTag. I would share this link directly, but as of today I can no longer find it in my history. As a result, I've uploaded the program itself to Amazon Cloud Drive here. If the author of the program has any issue with my re-hosting of his work, he should get into contact with me; as it stands, I have done my best to contact him and received no response.
This version of the program works on mp3, m4a and mp4 files. However, attempting to use it to modify the metadata on a file larger than 4 GB results in the file being corrupted. Yes, this means that if you do not have a backup, the file will be permanently destroyed!
Furthermore, the ability to select and change the properties of multiple files at once is completely broken in this version. There is an option at the top of the GUI to select multiple files, but only the first file is actually modified, leading me to conclude that this is a bug rather than an unimplemented feature.
The other Mac port of EasyTag (which I linked to in my initial question) is able to modify multiple files at once, but as stated in my initial post, it only works on mp3s. (I don't have any mp3's larger than 4 GB for obvious reasons, so I have not tested whether this functionality is present)
The other program which I've found is mp4box. Despite being a command line utility, it is able to modify the metadata on even very large m4a and mp4 files, and BECAUSE it's a command line utility, it's easy enough to construct a for loop that will let you modify multiple files at once. However, as you might expect from a program called mp4box, it doesn't work on mp3 files--and trying to do so anyway will leave you with corrupted data, which, again, means that you'd better have a backup in case you make a mistake.
So to conclude, we have:
1) EasyTag Port 1: Modify metadata on mp3, m4a, and mp4 files smaller than 4 GB. Does NOT allow batch processing.
2) EasyTag Port 2: Batch processing on mp3 files. Does NOT work on mp4 or m4a files.
3) Mp4box: Batch processing of m4a and mp4 files of any size, including those greater than 4 GB. Does NOT work on mp3 files and does NOT have a GUI.
Used together, these three programs support all of my use cases. What I cannot find, however, is a single, stable, and reliable solution.