There are a few ways to go with this... If you are looking for the cheapest way periord, you could weld the spider gears in the rear, but it will be horrible driving on the road around town... If you are looking for an inexpensive, but practical solution, you could do a Lock-Rite or similar "lunch box" automatic locker. These lockers just replace the spider gears inside the differential and can be done in an afternoon by anyone that can follow directions and has turned a wrench before. The first time I even SAW the spider gears in my differential was when I pulled the cover to install my locker. If you are a qualified mechanic you can go to the "full carrier" lockers (Detroit Locker, ARB, etc.) but it sounds like you are on a budget, so I won't talk about those.
As far as whether it will hold up or not, it really depends on you, the driver. I had a Lock-Rite in a YJ with a 35c rear axle and 32" tires for 2 years and didn't have any problems with it. At the end of the 2 years I did have a bent axle shaft, but I think that was more due to the 32" tires than the locker. I didn't do any rock crawling with that Jeep, just mild trails and some mud and snow.
If I were you, I would go with some sort of "lunch box" locker (see above) and start saving for some new Superior axle shafts for your rear axle. If you are only going to run 31" tires, you should never have any problems with this setup as long as you run the Jeep.
Good luck with your decision!