What happens if your iPhone becomes damaged while preparing for AppleCare+ Repair?
I would send it in, protect it as best you can - write a short note, I tried to get the SIM out, that didn’t go well, please help.
Send it off. If they call you and requote, you can decide if you pay more. If they fix it all, you’re set.
If you can’t write a short note, maybe just send it, but I would expect they’ll wonder more if it comes in shape not the same.
If you really are concerned, maybe get photos and call - but I’m guessing Apple will say send it in so we can look at it.