I have owned and still own a few GMT400 and GMT800 trucks both 1/2 and 3/4. Your trans should not be getting as hot as it is with the weights you are talking about and 4.10. These are my guesses as to your issue, numbered from most to least likely in my opinion:
1. Your driving style kept RPM lower than designed and therefore your converter was unlocked and you were out of torque curve. Up hills with that load and 4.10 at low highway speeds you should be locked up in at the top of 3rd. The LS will rev more than your previous Vortec did, and transmission programming has changed to reflect that. The tow/haul mode does more than just trans shift points so be sure to use it.
2. At some point the radiator was replaced with a cheap aftermarket which notoriously have a very short straight piece of restrictive copper tubing in the tank as the "trans cooler" versus the large heat exchanger section that was in the tank of the OEM radiator. If this is the case then you are basically asking the external trans cooler to do all the work and it is fairly small. There are photo comparisons online of cut-open Autozone-class replacement radiators vs. OEM class and the biggest difference is usually the ATF cooler loop inside one end. Chevy designed for significant trans cooling to happen in the radiator (more than 50%.)
3. The trans is tired or has internal issues and needs rebuilt/replaced. Luckily this is a common transmission fairly inexpensive to rebuild/replace and the GMT800 trucks are worth it, especially the 3/4T.
Just my opinion/guess
