Thank you so much, everyone! He was acting totally normally, but just in case I took him into the vet. I’m a relatively new bunny parent (4 months) so more for peace as mind.
The vet took a look at it and said indeed the nail had broken off (you could still see some encrusted blood), but she said no need to clean it or do anything, and that he’d grow it back. She said it looks like his nails had been short enough, may have been unlucky and got it caught on the rug when he was trying to jump away.
Also they let me practice picking him up a few times using the towel method. He hates being picked up, but I was trying to try to do it more (i.e. once a week or so) to get him used to it. The vet said the regular method I was shown by the shelter people (i.e. hand in between the legs, then other hand to support the butt) was challenging, especially given he is squirmy.
The method the vet showed me seemed to work. Throwing a huge towel on him, facing him towards you (so he won’t be as likely to run), covering his eyes and then wrapping one hand around his neck and the other supporting his butt.
I have renewed confidence in picking him up — I was going to avoid picking him up in the future given this experience, but hopefully the towel method works!
In other news, he gained 0.3lbs (from 4.3 to 4.6) in <3 months. Vet said still normal but would advise reducing his pellets.