1st off, in.
Next, I know you dont want to clutter your training with assistance but I think it might help you...
Hook grip, painful at first but you get used to it. DONT DO IT FOR REPS, you will rip off your thumb skin and have to start all over. Maybe just use it for your warm ups and first set of 5. Then straps. Train it by using (on everything) and with static holds and plate pinches etc.
*LIGHT WEIGHT!!!! YEAAA BUDDY!!!!
I was going to ask if stopping the workout was an option? And this is where some dumbell bench and military pressing would probably be very good assistance work.
Another option would be to on light day instead of benching do Dumbell Bench and Dumbell Military Pressing?
As for back imbalances are you using a mixed grip on deads? Do you sit at a computer a lot? Or use other unbalanced positions habitually? And dumbell power rows could help you develop evenness if you think you have imbalances. (Even barbell power rows should help too)
Some shoulder mobility work could also pay dividends. But if you are doing some of those things that cause imbalances no matter how much work you do to correct it they will still be there as long as the cause is still being performed.
May the progress be with you.