I’m going to answer outside of the context of 5/3/1, since I originally saw this in the BSL forum.
I don’t have an answer for you, but maybe some information to help make your decision.
I did a month where I squatted 6 days a week, 5 of them heavy, 1 medium. During that time I also researched the hell out various protocols that involved near-daily squatting. But this could apply to other lifts too.
Since I squatted to daily maxes, I had a decent feel for how my strength fluctuated.
The setup was:
Tuesday: heavy, then do overload work (limited ROM pin squats)
Thursday: heavy, overload work like Tuesday
Sunday: rest, or some other light lifting (not squats)
There was no real change in volume between, it was just working up to a daily 1RM, and some backoff sets. The intensity fluctuated according to what I could handle, but not by any external calculation or whatever. So you can kind of say that everything was held constant.
What I found was that Mondays were my strongest days, and Thursdays were my second strongest days. The lighter weekend allowed for some recovery so that I could hit PRs every Monday. Then I think the Thursdays were because of the overloaded work on Tuesday getting adapted to 48 hours later.
However, this approach also relied on accumulating fatigue over the course of the weeks. I was more run down as each week progressed, then the 4th week was a light week, and at the end of that week I set a fairly significant PR. The weekends were “mini” recovery periods, even though I was still pushed down, then the last week was a true recovery period where I was able to take advantage of all the gains.
Long story short… doing it more than once a week can lead to some improvement, but be careful of that “second layer” of fatigue, and make sure that whatever you do that you allow yourself to recover from that. If you don’t, you might make good progress for several weeks, and then hit a serious wall.
Even shorter explanation, 5/3/1 was designed like it was for a reason. If you tweak with the weekly loading and volume, you’re not really doing the program and you’re moving into untested waters. Managing long-term recovery is going to be your biggest concern this way.
EDIT: I did get a ton of really good practice with the squat this way. I wouldn’t recommend doing what I did, but there’s definitely value in just getting in lots of squats under the bar. Maybe just do some 50% squats after every session focusing on the same form as your heavy sets… light enough to have no real effect other than practice.