I'm pretty sure Weird Al is actually using compulsory licensing in music and just paying the required royalties to the songwriters. Anyone can cover any published song, you just have to pay the royalties when you do.

He doesn't actually make very heavy use of the satire plank of fair use. He credits the original artists. From his own website

"Does Al get permission to do his parodies?

Al does get permission from the original writers of the songs that he parodies. While the law supports his ability to parody without permission, he feels it’s important to maintain the relationships that he’s built with artists and writers over the years. Plus, Al wants to make sure that he gets his songwriter credit (as writer of new lyrics) as well as his rightful share of the royalties."

The fact that he could rely on fair use is separate from whether he as an artist does rely on fair use.

Weird Al makes a point of getting permission even though he views his satirical works as fair use.

The "commercial" test is only a part if the criteria and not necessary, but to say it has little impact is clearly false.

Always great to see people point out Weird Al, cause he's the shining beacon of an example of what OpenAI et al. should be doing. He explicitly gets permission from the original authors before doing any of his parodies, and he's even been turned down a few times as well, famously Prince rejected him a bunch of times and he subsequently has never made a Prince parody.

Not only does he get permission from the original authors, he also pays royalties to the authors despite legally not having to do so.