31x10.5R16 typically measure 31" dia. x 10.5" wide, 651 revs/mile.
245/75R16 typically measure 30.5" dia. x 9.6" wide, 662 revs/mile.
265/70R16 typically measure 30.6" dia. x 10.4" wide, 659 revs/mile.
265/75R16 typically measure 31.6" x 10.4" wide, 638 revs/mile.
All will fit on stock Moab wheels.
None should have any issues with rubbing.
If I were choosing a metric tire to replace a 31x10.50R16 I would probably choose a LT265/75R16 for the extra height, but if you don't want any extra height a LT265/70R16 would be the closest to the tire size you have now.
There are many non-Jeep-brand overlanders who run LT265/70R16 and LT265/75R16 tires on their rigs with success.
I have a set of Cooper Discoverer AT3 tires on my Xterra, LT265/75R16-C, that recently completed a month long overlanding trip to Cabo San Lucas with zero issues. I recommend them and I'm sure they are available in 265/70R16. [They replaced a set of same size Cooper Discoverer ST/Maxx which I also highly recommend even though they are load range E.]
YMMV