I would plan on calling the vet in the morning, but I wouldn't fret too much. I would be kind of surprised if your rabbit swallowed a piece big enough to cause problems. Rabbits normally chew their food really well so swallowing something large would be uncomfortable and unnatural to him. In fact, it is pretty scary to think of all the bits of carpet, cord, baseboard, shoe, etc that have come up missing after leaving a rabbit unsupervised. lol. But they are little destroyers and even if they do swallow that stuff, it is usually tiny pieces that are of little consequence to their systems in small amounts (provided there is nothing toxic ingested, of course).
And their systems are designed to handle roughage like hay, so indigestible matter is nothing new. My only concern would be if he bit it off in such a way that there is a sharp or jagged point that might perforate him internally.
If it were me, I would completely clean out his cage or litter box tonight, and monitor his pee and poop. If it keeps coming, that is good.
Another thing I would do is give some petromalt. I don't know if you have it on hand, but you will never catch me without it. It is a hairball remedy for cats, but if you get the malt flavor, the rabbits just LOVE it. It essentially lubricates the inside to help stuff pass and prevent blockage. If he did swallow a larger chunk, this would surely help it slide through easier.
Keep an eye on him to check for signs of pain or discomfort. If he flops on his belly in a way that is not normal for him, watch him and see if he gets up and lays down repeatedly. That would be a sign of belly discomfort.
He absolutely should see a vet asap if he acts uncomfortable.
If he grinds his teeth intensely then he is having major pain and you probably need to take him to an emergency vet.
If 24 hours passes and he is still peeing, pooping, eating and hopping around like normal, then you are probably in the clear.
But regardless of how normal he seems, I would call the vet as soon as they open and get their advice directly. If you bring in the syringe then the vet might be able to tell how big a piece is missing and can make a judgement then.
I am not a vet. I just have a rabbit. I have seen him through a few health scares. This is just what I would do if I were in your place. If anybody disagrees with any of this advice, then please correct me. I am open to learning as well.
