Hmm, I have no idea whether sizes of batteries are constant between countries or if different versions of Loopin' Louie use different batteries but I bought a (probably fairly old) copy of this game from a 2nd hand shop about 2 weeks ago and it takes a C battery. In Australia AA are skinny and C are fat (and D fatter still).
I can only answer the battery part of that question-- In the US, the AA, C, and D batteries match your description and I've purchased batteries in Europe that worked the same way as well.
The question about different versions of the game is pretty critical to the birthday boy, that's for sure.
The version sold on Firebox U.K. lists AA batteries.