well, IMHO, when it comes to tires, cheap and good don't go together. it's not the lift, lights, or axles...it's the tires that make a Jeep. to me, it's the single most important modification, not so much height, but tread and quality.
traction is the name of the game off-road, as I'm sure you're figuring out. best thing for traction issues is an upgrade to an aggressive tread pattern. the good ones, aren't cheap. opinions will vary, but I place Pitbulls, Irok's, and Swampers as my "high level" tires. they cost a bit, wear like sh*t, but they climb up the sides of a house.
you could cheap out a bit and get "good" tires. Mickey Thompson MTZ (I currently run these), KM2, or Maxxis Buckshots all handle deep mud well, and they don't wear too bad. if wear isn't an issue as in your case, I'd probably stick with my first group.
last is my iffy group. Mud Rovers, BFG AT's, Duratracs, etc. not bad tires mind you. but rocks and mud approaching 12" they just can't hang. the price is attractive, and they wear well, but offroad they aren't quite there with the big dogs.
honestly if I had a mud toy jeep that sees only offroad duty, I'd get some Irok's, Groundhawgs, TSL's, or even directional tractor tires. even my tires, although very good, is simply a cheap imitation compared to those. they are the big boys. you want max traction, there's your answer. not cheap, but you need to find that balance of $$$ vs. traction.
since you see no street, rule an AT out. get a KM2/MTZ/MTR radial AT LEAST. but I'd really consider a Pitbull/TSL/Super Swamper style instead.