Depends on the size and shape of the piece. You will seriously luck out if he regurgitates it. If you see it protruding from his tummy, I'd get the fish out and gently massage it out towards the throat while gently sticking a finger or two down his throat - keep the fish in water in a shallow container. If he does not regurgitate by himself, you may be able to grab it and pull out. If the stone is very small, he may pass it but it will take some time (he may refuse to eat for weeks) and it's a gamble.
I'm speaking from my experience and that of my friends and what I saw on YouTube.
The stress of catching often make fish regurgitate too on their own. Hence, I'd not anesthetize it.