Rory,
Well if your gym is closed and push ups are your only option and their not causing an increase in pain or symptoms then continue. However, this may come back when you return to the gym. Below are some common faults I see with this issue.
Shoulder asymmetry is fairly common issue based on age, injury history, dominant hand etc. Most likely the asymmetry is not the problem, its the way you support the bar on your back and arms that's the problem. When you get under the bar and set up, a majority of the weight needs to be supported by your back (in the low bar position) and very little in your hands. In fact, your hands are just laying on the bar compressing the bar towards your body, not supporting from underneath. I would make sure that you address this by recording yourself and reviewing your videos.
Regarding the bench and time away from the gym your going to have to reset anyway. So I would reset with a conservative, pain free weight (may even be the empty bar) and avoid the lockout position early on. Start by completing 3 sets of 12 reps on session one, then add 5# and complete 3 sets of 11-12 reps on session 2, then add 5# and complete 3 sets of 10-11 reps on session 3 and so on until you return to a 3 sets of 5 and a full lockout position. This will take 4-5 weeks to complete.
Please be advised that its typically the handling of the bar on the back thats causing this type of tendinopathy and is exacerbated by the bench. So you must clean up the squat first.
Good luck