I don't think it is possible to fix a small hole in a shot gun barrel. The walls are thin to start with and if the barrel is corroded there is likely to be other weak spots.
I think the only way the gun could be brought back into to use is by re-sleeving it. To do that a gunmaker would have to cut the barrels off just forward of the breech and braze in a new pair of barrels. Then it would have to be reproofed. That is very expensive and probably not worthwhile for the gun you have.
https://www.vintageguns.co.uk/articles/barrel-lining/