I saw an orthopedic surgeon a couple of years back for a similar sounding problem. I have had mobility /impingement problems in my right shoulder (at least) since I broke my collarbone in 2005. I started playing tennis, and then started weight training (5x5) and my shoulder got pretty painful. I could move it in all the directions, it just hurt when I played tennis, washed my hair, etc.
The doctor I saw gave me the push tests and said it definitely wasn’t a complete tear of the rotator cuff. He said the x-rays looked fine, and there was no point doing an MRI to see if it was a minor RC tear or tendonitis, because the treatment for either would be the same. 2 weeks of anti-inflammatories (naproxen) to calm it down, and about 6 weeks of PT. No sleeping on my right side, no tennis or weights for a month.
After a month I got back to tennis with an extended warm-up routine, and cool down stretches to do after.
I was planning to get back to the gym, but I never got around to it before Covid happened. For most of 2020/21 I did a few kettlebell exercises and a bodyweight program from GMB. I was pleasantly surprised that the GMB program seemed to make my shoulder feel better rather than worse. I have been riding my bike and playing a little bit of tennis and I haven’t really thought about it at all.
I started back at the gym a few weeks ago. I’m using dumbells for overhead press and being very conservative with the weight progression, but my shoulder already feels slightly less happy, so we’ll see how far that goes…
Hope this gives you some hope. BTW, I definitely don’t have hyper mobility.