I've seen "Which Hand" performed as a bar bet, but it has also been performed as a Mentalism Effect. Which category do you think it deserves to be in?

It all depends on presentation. If presenting as a bar bet, it won't come across as a mental trick, in the same aspect, if presenting as a mental theme, a bar bet would be redundant.

I suppose performed as a bar bet gives the impression that it's a 50-50 bet.
Whilst performed as a mentalism effect suggests there is "magic, or trickery" behind it.

Just my thoughts ... could be wrong

Performed well there is an element of mentalism, say the hand silently is n your mind not out loud, but equally it can be performed a bar bet.
Hey I may give this a go for drinks I’m usually pretty good...

This is entirely down to scripting how you present it.