Several things have helped me develop and maintain a bodyweight press, even as my bodyweight has increased.
The first thing I would say, and it appears you have already done this, is make the press a priority. Two or three years ago I set out with the goal of a bodyweight press and ended up pressing my bodyweight for a triple (at 195 lbs). I believe I was following 5/3/1 at the time. Just a few days ago I pressed 210 lbs at a bodyweight of 205. That seems like a back track from a few years ago, but pressing has kind of taken a backseat to to other things, even though I still love it. I’m currently doing 5/3/1 again.
I’m a natural presser and it has always been my favorite lift, but there are a few that I did that really seemed to help. Switching to a false grip (or suicide grip) really seemed to help the bar track better. I know a lot of people think it’s dangerous, but I personally have never had an issue with it (knock on wood). I even use it in the bench press. Later on I heard the same advice from Wendler himself, so if that lends any credence to it there ya go.
I see a lot of guys that press with an extremely wide grip. When I press my grip is so that in the bottom position my forearms are perpendicular to the floor (this is narrower than a lot of the guys I see). Also, in the bottom position I am actively pushing my elbows forward. If you sit at your computer and act like you’re at the bottom of the press with your hands, move your elbows back and forward and feel the difference. With elbows forward your hands start to move up because your lats engage and form a very solid base to push off of. This may be the most important thing for me in the progression of my pressing.
The elbows forward thing leads into the next thing. Train your back religiously. I don’t know your level of strength, but I do about 5-10 chins between every pressing movement (except for the last two sets generally). Sometimes I change it up and do facepulls, but it’s usually chins or pullups. If you can’t do that many, do 1, 2 or whatever. Just do not go to failure or near failure. You don’t want it interfering with your pressing. This just helps get a pump into your lats and helps activate your lats so that you have a better base to push off of. A nice perk is it adds strength and size to your back. An alternative if you’re weak in that arena is very light wide grip pulldowns for higher reps really concentrating on activating and feeling your lats. I did this myself and it helped immensely.
As far as directly training the press, a progression like 5/3/1 is great, in my opinion. After my top set I always drop down and do a back off set or 2 for 10-12 reps. I’ve noticed that for me, volume is the key to increasing my press. This really comes into play for me when I get closer to my true 1 RM in 5/3/1. I essentially do “first set last” from Wendler’s new book, but I was doing that before he ever put a name on it.
Sorry for the long read. I haven’t posted on here for years, so maybe I’m making up for lost time.