Hey Smont,
Many people (myself included) can bench with good (not great) technique and even a few muscle imbalances when using moderate weight (reps at 8 and above even taken to failure) and not feel anything out of sorts. Heavy weight can result in strain (slight pull can be from minor strain) if form isn't locked in or certain muscle groups are less developed than others (ex. underdeveloped back compared with pecs/delts/etc.). Lots of muscle fibers firing quickly with heavy weight and lower weight early the set. I have a few thoughts.
Get tight on your heavy sets. You'll often hear coaches say to powerlifters 'get tight' or 'stay tight'. That que works if the lifter knows all the things that need to happen to 'get tight' but it takes practice. Tucked shoulder blades, pressing traps into the pad with legs, and controlling the weight while lowering are important to being 'tight'.
Get a good chiropractor. There are a lot of bad ones and they can cause more harm than good. Sometimes something as simple as getting adjusted can solve the issue. A good chiro works wonders. My chiro can guess what lift causes pain after a quick scan. He give advice on how to adjust form and supplemental exercises. That's what you want. Somebody who can give you things to work on outside of their office not just "come back next week".
Strengthen your spinal erectors. If you don't already do them add it hyperextensions (even better if you can do them with weight). I do 4 sets of hypers twice per week (back and leg day). I'll grab 2 25 pound plates and do 25+ reps for 4 sets. I drop the plates on the final set and keep repping until failure. This has alleviated low back pain from sitting too much. I have short arms but still pull 650. My back is still in one piece because of all the weighted hypers. My erector pop out next to my spine like steel cables. Leverages suck and I NEED a rock solid low back to overcome that when pulling.
Pull more than you press. I row twice as often as I bench. Punish those lats they can take it! Big lats equal a wider more sturdy base to bench press from. Double up on pull ups, chin ups, and pull downs. I used to do them once a week but that left gains on the table. I like heavy yates rows and lots of other rowing movements. Check out this video for my preferred rows. youtu.be/ktiZpJTbLRk
Do targeted glute work. I love glute bridges! I used to think this was for the girls. Man was I wrong. Targeting glutes with weighted exercises enables you to flex the hell out of them for extended periods on heavier sets. Gotta flex the butt while heavy pressing. Jump to 3:30 to see how I like to do them. youtu.be/2p_MghDB6Do
Most people use heavy weight and do low reps but you're better off hitting higher reps (20+) with a lighter weight so you feel your buns burning like hell on the last 4-5 reps. I do this on leg day after squats. This will have great carry over to bench and help reduce back strain. On the heavy sets using leg drive to press your upper traps into the pad while SQUEEZING the heck out of your glutes the whole time protects the low back. Game changer for a lot of powerlifter.
When I benched 405 I thought my form was great! It sucked and eventually got injured. Now I'm the same bodyweight a bit stronger but MUCH better form and approaching 500 injury free. Nothing feels off and it's awesome.
I recommend avoiding going to failure (stop 2 reps short) while the shoulder is sore. Stick to lighter weight and maybe even dumbbells until it's at least 90%. I agee with your thought about overcompensating. I backed off heavy pressing for a year then built back stronger with better form when my shoulder was 100%. Muscle memory is a very real thing. I actually loved training bench again at lighter weights after taking a full year off from it. Adding weight to the bar every couple weeks rocks. You know what aggravates it... don't do it lol. I'm guilty of working through pain when I know better because I love certain exercises (cough cough bench press).
I do form checks for 10 bucks if you ever want me to take a look let me know. I literally spend an hour or more to review the lift. If you're wondering what my background is I'm a board certified behavior analyst with a health ed. degree. I have a state record for bench. I've been lifting for 29 years. I coach a few advanced lifters but my time is unfortunately limited. I have 2 boys and work in educational administration. Let me know if you have any other questions.