That's what I'd do too. My shop would charge 20.00 for the troubleshooting. If any parts were needed we'd absorb that 20 and just charge whatever the repair cost is. Look for a shop that does that and also preferable no solution no charge.
I personally haven’t tried replacing the rear camera lens on this model but I can’t imagine it’d be more difficult than installing it into the bracket after cleaning all previous adhesive from it. I can tell you from experience its most common that you do not have to remove the back or any other component from the phone to replace that lens.
When you mentioned the screen wouldn’t close was that the old battery or new one? To me this issue sounds like 1 of 3 things. You could have a charging port issue but that’s not entirely likely since it does register as being plugged in. You could have been sent a defective battery and in that case would need to replace it. Your phone could have a Tristar issue. Out of those three I’d say its probably a defective battery
If you’re talking about the side-frame screws that hold the LCD shield in place the can be snapped off carefully. Is it ideal? No, but it can be done. Or you can purchase another backplate and be done with it.