Need to know which LT1 are we talking about. The original 370 horse solid lifter four bolt main LT-1 from 1970-74, the LT1 reverse cooled version with an Opti-Spark distributor sticking out of the front of the motor attached to the cam in a one piece rear main seal block, or the 1996 Gen III all aluminum small block LT1 that is a computer controlled fuel injected only version of the motor.
If you have the original optional LT-1 (with a hyphen) engine that was introduced in the Corvette in 1970 I would definitely rebuild it.
It was a raucous, racing-inspired engine with solid valve lifters, a high-lift camshaft, and high-winding high performance small block. Displacing 350 ci (5.7-liter) with a compression ratio of 11:1, it was rated at 370 hp at 6,000 rpm and 380 lb-ft of torque at 4,000 rpm. It was enough to push the Corvette from 0-to-60 mph in less than 6 seconds. A true high-performance powerhouse, featuring many heavy-duty components not found on the Corvette's standard 350 engine, including a stronger cylinder block with four-bolt main caps, forged aluminum pistons, a high-rise aluminum intake manifold, a baffled oil pan for reduced windage, a transistorized ignition system, and high-volume oil and fuel pumps. Only 1,287 Corvettes left the factory in 1970 with the LT-1 engine.
The rear end under the 1996 full size could be an 8.5 inch or a 7.6 inch ring version of the corporate rear end. If it was an Impala SS it will be the stronger 8.5 inch rear with 30 spline axles. In either case the mounting system was different as it had triangulated control arms (the two uppers turned inward to eliminate the use of a Panhard bar) and the springs shocks and control arms mounts differed from your car. They can all be cut off and rewelded if you are careful not to burn through or warp the tubes with too much heat. Finally your axle is 64 inches wide and from memory the '96 was only 62 inches wide. If so you will need new rear wheels to clear the rear disc brakes and account for the change in off set for it being narrower.
If the budget was a concern and that Caprice LT1 was running well or low-ish miles, I'd say to go ahead and keep it until it dies then replace it or upgrade to something better. It's just convenient that you have it. It's definitely not in the top 3-5 that I would go with.
Like Dave said, that rear-end is going to need some fabrication work, and could induce some extra wheel costs to make everything play nice together. In my humble opinion, I don't think I'd try to use it.