Hey Matto,
I'd definitely bring a prime with me to those places, its very dim in there and i had to push my ISO up on my 40D to 3200 even on my 50 1.4 (i wish i had my 30 1.4 during this time [bought in hong kong same trip]) - but managed to shoot a few with my uwa (including below).
As for photography in the Daibutsuden pavillion, its allowed (well i didn't see otherwise - just no group photos) its a major tourist spot (for the locals) - but you'll need to jostle for a decent spot . Its very dim in there, and with the 50 it was a bit long (mind you it was a crop factor 1.6 for me) so i which i had the 30 for ease. Heck i even saw flash photography in there...
The below is with the 10-22 @ 22mm . inside the pavillion. can get you EXIF data if you like too.
So to answer your question. Go a prime, if budget is an issue - grab a 50 1.8 and get some lovely low light shots, or even the 40 pancake.