Your replies have been very helpful, thanks a ton.
I have been keeping a very close eye on him since the incident. I moved him up to my room (separated from my other bunnies who live in here) and have not noticed anything unusual. I will definitely be telling the vet about it and ask her to listen closely to his lungs. He's been eating and drinking fine and has not had anything similar happen again so far.
I might try again with the x-pen when I move him back down to the basement. I have also heard of attaching a sheet or light blanket to the top using binder clips. It's not a huge deal if he gets out because the room he's in has nothing really harmful to him, it's just that I'd rather him not get stuck under the bed, pee on the floor or decide to chew up all of the boxes and furniture stored down there.
When I first saw him I also thought New Zealand but now I'm at like 70% that he's an American White. His posture is mostly what makes me think that. When I pet him and his front half lowers to the ground, he has a certain downward curve in his shoulder area. Here's a photo I took of him today:
Although I'm really not the best with larger breeds and have only ever personally owned Lionheads. He's a very friendly guy and his fur is soft and somewhat silky. He weighs probably 9 or 10 pounds; he's almost the size of my cat