Why didn't they split the money up?

I disagree. Sick Boy certainly would have done it -- Renton even says SB would have done exactly what he did if he had thought of it first. But Welsh wrote the character Begbie as a deranged psychopath who terrorizes his friends but also has his own twisted sense of ethics and believes strongly in his own brand of loyalty.

I agree with you. Begbie was the one protecting the money. I bet he also was planning on taking all of it. Just waiting for the right moment to do it. My guess is he would wait for someone to do something stupid, turn that into an argument, say that idiotic thing was justification for him to take the money, and threaten to kill them if anyone disagreed. In that situation not one of them would have stopped him and he knew it. He just needed the right moment so he can build up whatever reason would feel right to him. He feels superior to all of them so he needs to invent some reason to take the money no matter how flimsy.

As for the bank, Begbie is a wanted man. He could not make a bet earlier in the movie. He has to be careful about where he goes and what he does. A bank teller will probably not recognize him but there is a chance. Besides, I bet Begbie never had a bank account in his life. He is on the run and will use the money to pay for cheap places to stay and to keep drunk. Try to find people to bully into taking care of him.