You can easily run 31x10.5 on your stock wheels without a lift. You may possibly get a rub on full turn. (I did not) Not until I put on my lift. Now I get a little rub on the lower control arm at full right turn.
The height is not a problem. I have a 2-1/2" lift with 31s and park it in my garage with a 7' door.
Here are my progress pics.
31x10.5r15 TreadWright MT Ultras on stock wheels
2-1/2" Rough Country lift
Sorry, I just like to show it off.