I am doing an LS1 CJ7 build and here's my opinion after many months of deliberation. My CJ will spend most of it's life on the road, but I want it to be capable when I do take it off road.
Rear axle: With an LS1 the CJ needs a strong rear axle during daily driving anyways. This way you can actually drive it like it has an LS1 in it when you really want to and not worry about breaking anything. Without a strong rear axle you're shorting yourself because you don't get the full potential of the Jeep. I picked up a Ford 8.8 from a salvage yard for $150. You can get the axle flanges redrilled to 5x5.5 for $60, buy some new spring perches ($30) and now you have a decent axle that will not break the bank ($240 total). Also, this axle is about 59" WMS which is about wider than the CJ widetrack (about the same as a YJ). This will give you better handling and road manners, but you'll want to change your front axle as well.
Front axle: Not as important as the rear on the road, but off road it isn't even in the same ballpark so it'll be the first thing snapping when you hit the trail. Dana 44 is the answer. Out of all the options the two that are the most cost effective is finding a Wagoneer front for about $300-$400 or cutting down a Dana 44 from a late 70s Chevy truck/blazer. You can find a Chevy dana 44 for about $100 and there's one for every square mile in the US. The waggy option is harder to find and the chevy option is more involved. With the Chevy option you also have to change the lug pattern from 6 to 5x5.5. I can give you more details through pm, but can be done for less than $500 total (including the axle).
This setup also gives you 4 wheel disc brakes which will help stop your speeding bullet. So for less than $800 you can do the axles and that leaves you $1700 for other stuff like ARBs, stronger driveshafts, YJ spring conversion, power brake conversion, etc.
A great paint job would be nice, but it's a 30 year old vehicle that you are not going to be taking to the show. Think about how content you will be being able to take your jeep offroad without it breaking at the first sight of a boulder.