You can fit the 31" tires on the stock rims with no lift.
As for the tires, they are too very different mud terrains, and your purchase depends on what you want to do with them.
The BFG MT's are a harder rubber compound, you can get about 50K miles out of them. They do well in the mud and pretty good in most all conditions.
The MT/R's are a much softer rubber compound, you'll be lucky to get 30K out of them, I've seen one set last only 12K on a TJ. The softer compound makes them great on the rocks, super sticky, but they're only marginal in the mud from what I've seen. They don't self-clean as well as the BFG's.
The MT/R's are more expensive than the BFG, so is it worth it to pay more for a tire that will wear down quicker? Maybe to some, but not me. The MT/R hit the market when the market was hungry for a new MT tire, and it's been a fad ever since. To me, it's overpriced and overrated.
If I bought new today, I'd have to go with what I could get my money's worth out of, not what was popular. I think my choice is evident.
The flame suit's on for all you MT/R lovers, fire away :lol: