Aye, he does accept them - he's taken every one I've ever offered him, the '2 types' thing above notwithstanding.
However, I DID find that occasionally he grew bored with serving the same customer over and over, and I had to wander off the screen, and then come back again, before he would give me any more.
He should take account transfer and advanced character tokens quite happily.
If he doesn't, and you have a fair amount of time on your hands, I'd suggest calling a GM. There are lots of things you can do at a bank while waiting for GMs!
*insert list of crazy things to do at a bank below this post*