I wouldn't run stickies on it in it's current setup, but that's just me. Chad (zukimaster) went from 36 tsl's on his Samurai with built Toyota axles and beating the absolute dog **** out of it never breaking, to 37" red labels and could look at a rock wrong and it would break. I remember very few times him going wheeling and not coming home with something broke after he switched to reds. Now what's he doing, going to tons and building a whole new rig haha.
Now a set of regular TSL's that are brand new will suprise you how good they will grip things but still give enough to not be so unrelenting as a set of Reds. When Cole Blake used to wheel with us at Iron Gap, I remember him showing up one day with a brand new set of 38.5 TSL's on his Toyota crawler on leaf springs. I was amazed at how well they gripped on those rock ledges in the creek beds. Regular TSL's are a good happy medium tire, decent grip and easier on parts than Reds.
Go with tons! Go with a lower geared t case! Go with new tires! Sell it and buy a buggy!
Realistically there is no cheap solution to your problem, thus the name of the game and what I don't miss about the sport. I finally had my last crawler pretty solid before I sold it but little stuff still tore up on it here and there like a clutch master cylinder and other random ****. I won't get back into wheeling until I can afford a rig that is virtually bullet proofed from bumper to bumper.
If you wanted to set a goal for your current rig, you need: tons, stretch it, bigger tires, better transfer case setup to allow lower gear (atlas, doubler, etc - not real sure of the t-case options for Jeeps), and you would be set. You could put just tons under it now but it would still suck balls trying to crawl obstacles with a stock t case. There is no one fix-all for your problem, if there was it still wouldn't be cheap haha. Good luck!