I would definitely get him checked with the vet for possible parasites, as he is a rescue, and you have no idea of his history. This could maybe be done using a stool sample, but see what your vet suggests,
Also if he scoffs his food very quickly without much chewing he could be swallowing air and that would cause much more burping.
Dogs can be quite horrible eaters. If he is fed whole prey, there is a high chance he is swallowing very large chunks of it, bones and all, without much chewing, if any. They are hard-wired from their wild roots it seems, to be able to do that in order to cram down as much food as possible so some other creature or pack member won't get it!
But it can play havoc with their digestion
Also if he is a rescue dog he might have picked up a "wolf it down" habit because food was scarce for him (?)
Just as it is with us, saliva and chewing are the first stages of digestion.
Or it may be because there is something he is finding indigestible in his diet. I doubt if a bit of pizza crust would cause it (unless he was fine until he ate that?)
Of course it's not "healthy" for dogs to eat pizza crust often, but a bit occasionally won't do any harm usually. It's basically only flour and water with a bit of oil and dogs as omnivores can usually handle that on the odd occasion.
I would say -speak to your vet but in the meantime you could perhaps run his food through a mincer and see if that makes any difference?