Thanks for chiming back in on budget.
This is my advice and probably last post: Doing an engine swap is going to take time. Are you doing the work yourself? You need to do a lot of research on topic. The good news is that LS swaps have been swapped in just about anything, and the GMT400 is probably the most swapped truck they go into. Maybe you can drive the 2020 until yours is back up if you decide on it.
I would highly recommend a LQ4/4L80e swap. You won't be disappointed. Get it as a combo with ECU and all accessories. They usually go for around $2500 here, but sometimes you can get good deals. I saw one the other day for $1500! You'll need to add another $1,000 to $1,500 for miscellaneous things to complete the swap though. There's different way to do things on the wiring/electronics.
My mason has a 2500HD with the 6.0 and dude works it hard, towing brick, CMU, sand, etc. Just about all of the older work trucks in construction and landscaping around here pulling trailers are GMT800/900 2500HDs, the majority of them with that combination. It's starting to become rare seeing any GMT400 8 lugs anymore as work trucks.
As I mentioned earlier, the Gen V has about the same power as the older LQ4, and the K2XX is heavier than the GMT400, so you can imagine it will be a good combo for your 97 since you have driven both. Reliability? Stay stock. A Gen III is pretty reliable. Not much to go wrong with them. 4L80e is a joy to drive. Pretty dang tough, reliable units. On a power/durability to dollar ratio, you can't beat the LS/4L80e.
Keeping your engine will have an advantage of truck not being down for upgrades. Headers, cam, and exhaust, etc. are pretty common upgrades. If you opt to punch out the block or open it up, then again truck will have downtime. Then you have to be weary of your 4L60e: they are not too reliable for working them hard on a daily basis. Throw some big power at them in a heavy truck towing regularly, and you'll need to upgrade sooner rather than later. It's gets spendy after a while rebuilding them.
I hope this helps and good luck. Whichever way you go, consider doing a truck build thread. Well all like those here.