MP3diags didn't say anything aside from some normalization information missing.
Tried every possibile transformation on mp3diags. Some transformation did work. Had to restart MusicBee to actually make it look like it was fixed.
Since at first (before restarting) the problem didn't seem fixed, I tried comparing the file and the transformed file with an Hex Editor.
I found that the alternate title was appearing just after the Lyrics of the song, in the file structure, and after the text "TIT2".

TIT2 is the name ID3v2.3 and ID3v2.4 use for the title tag. But there's more than one in the picture.
I went to the original file on MusicBee and tried with "Delete Lyrics and Redo Search". This fixed the problem temporarily! But after playing another song and then playing that original song again, the problem returned.
I am currently using the Beenius Plugin to retrieve lyrics.
The weird thing is that the problem also happens on songs for which I get no results from the automatic lyrics search.
In the past I was using LyricsReloaded to get lyrics from Genius, maybe the process also wrote some other information aside from lyrics into the file, and now it has to be manually deleted. I noticed that searching for the song as "Down Like Wow" on Genius actually works.
Very confusing stuff. This is probably the best I can do to help identify the problem.