Bowling doesn't - as far as I know - have any rules regarding how you throw the ball. (Bowling alleys certainly have rules, which is why he said he got kicked out once, because they thought he was goofing around. I would also imagine that lane operators wouldn't care too much for people throwing a ball overhand as that might damage the lane.)
A couple of the best bowlers in the world today use two hands and no thumb in the ball ... that's pretty unorthodox as well, but there certainly aren't any rules against it.
I would think that if he was good enough then he would be just fine. But unfortunately, owning a 278 high game with a 204 average on a house pattern isn't even remotely close to good enough. I would say if you wanted to parallel that to golf then it would be the equivalent of maybe 4 or 5 handicap.