I’ve got a few recommendations for you to give consideration. They will be in no specific order.
Can you bench warm up weight without issue? Don’t know how much you’re benching but lets say you’re doing 300+ can you pump out 25-30 reps of 135 without pain and discomfort? Is it only benching that causes issue? I’d recommend dropping the heavy and even medium weight training and push the reps if its just bench for a while. Blood flow while still keeping things moving without putting a ton of stress on the tendons/connective tissue/bursa. How about your close grip bench - does it hurt you at all? Another option would be to do bench press for reps and the ‘pump’ but use close grip to push yourself and move heavier weight if it doesn’t cause discomfort. And I firmly believe that hammering away at close grip bench carries over SO much to your traditional bench press. I have to admit I feel pretty good when I’m close gripping more for reps then everybody else is doing for a few on traditional bench.
If you genuinely think its a tendonitis/inflammation thing then you could contact your PCP (not angel dust dealer) and seek out a script for Methylprednisolone and do a burst protocol to knock down the inflammation. MP is a corticosteroid and very effective at relieving inflammation. It makes some excited/stimulated so taking early in day is preferred. If they won’t give it to you, you could order it from AIPCTshop and do it yourself. Typical protocol is with 4mg tablets starting at 8 tablets working down to one - 32/28/24/etc mg per day. Always taper down dose, stopping abruptly can cause issue’s according to medical warnings. My wife accidentally lied to me and told me I had 8mg tablets from the doctor when in fact the were 4mg so my last burst was actually 64/56/etc and I’m still alive. She even works in pharmacy lol.
Sam Feroce (sp) had some videos talking about how you should be able to row as much as you bench. I really found it eye opening and can honestly say that doubling down on my rowing has definitely helped me in many ways beyond the bench. How far away from rowing what you bench are you? If the divide is wide, you very likely are suffering muscular imbalances that will only get worse.
On the topic of muscular imbalances - if you’re able to see a massage therapist or a chiropractor who can perform myofascial release of trigger points and knows their shit back there identifying muscle issues then the information they could provide you can be invaluable. This is how I learned that my rhomboids were very underdeveloped compared to the rest of the muscles back there and that my lower traps weren’t functioning properly. A shit load of batwing rows and concentrating on face pull movement pattern, and lower trap activation on row machine and hanging from pullup bar have assisted greatly in getting these back in the game. Rear delts fall into this category as well - don’t forget about them because you can’t see them very well, they need to be hit even harder then front delt work or they get lost in the shuffle and they are important for shoulder balance.
Another tie in to these back muscles is your posture. Are your front delts/chest overdeveloped (goes back to row = bench) and rounding your shoulders forward in every day life? What about traps - are they pulling your shoulders up chronically or helping your shoulders to round? This leads to chronic shortening of the chest/delts which will actually pull the glenoid (head of the humerus) forward in the shoulder joint which can reduce usable space and generate inflammation from additional rubbing/pinching. This chronic shortening in the front also causes chronic lengthening in the back muscles. The muscles attached to your scapula will be constantly lengthened which puts them in a mechanical disadvantage. The fix is to get them activating better/strengthened in addition to stretching your pecs/delts habitually so that you can bring yourself back to the position your body was designed to be in.
Foam rolling your back to get it loosened up/hit hot spots of tension can help you get ready and allow you to get the thoracic spine mobilized and ready to flex in bench position. Fixing the muscle imbalances, motor pattern deficiencies is the only way to avoid repeated issues in the future. Antiinflammation medication, BPC, GH, etc can only fix the issue now. Have to get to the root cause so you can keep lifting and stay healthy.