5.3L will be fine. The 6.0L was a fuel sucker from the factory. A few tight-lipped guys seem to have unlocked a tune that at least keeps it proportional for its size, but in factory tune they burn more fuel than you would expect compared to the 4.8/5.3 versions.
5.3L is plenty, and since folks don't think the 5.3 is big enough, they're much cheaper than 6.0L. If you poke around, you should be able to find a decent 6.0L for not crazy money. I got a 6.0L LQ9 (the 345hp high-output version of the LQ4, found in Escalades, SS trucks, etc) with 80k, wiring and ECM for $1300. They had it pulled, palleted, and ready to load on my truck. I think what kept that one less expensive was the fact that the accessory drive was busted in the wreck, so it came with water pump and alternator, but no brackets or other accessories. Since I'm putting it in a car, I need a modified F-body drive anyway so it was no issue for me. The LQ9 has the 317 heads, which are basically LS6 heads with slightly larger chambers (since the LQ9 is 6.0 and the LS6 is 5.7).
But 5.3L truck motors are as plentiful as bacteria and more than enough for towing. You'll run out of truck capability before you run out of torque.
Your old 4L60 will physically bolt up to the LS, but there are two bolt holes that won't line up. You'll have 5 which is plenty. The problem you'll have is compatibility. The 5.3L ECM will need some custom hacking to get it to run the old 4L60E. I would either build a whopper 700r4 and program the ECM as a manual transmission version to ignore the transmission, or just get the 4L65E that was behind the 5.3L.
I will, however echo what Junkyard Dog and Kenny said: The 5.3L will be a better motor all around than an old-school 350, however, a 350 will be 80% as good for a tenth the work and money. Don't get me wrong... I'm all about taking what you have and making it what you want, but you're looking at investing 5 times what the truck is worth that can't be recouped if you take it out and sell. But I've received that advice a million times and ignored it because its what I want so please don't think I'm being critical.
Consider this: A good running Vortec 350 from a junkyard = $500. Take the vortec heads off, have them cleaned and magnafluxed and sell them for $200. Sell the intake for $100. Buy a set of $600 aftermarket vortec heads with thicker castings from Jegs/Summit. Buy a used aluminum 4-bbl vortec intake that someone took off a ZZ4 crate motor or a marine engine for $80. Find the TBI from a 454. (454 TBIs are like a 4-bbl version of the 350 TBI). Swap over the dizzy and accessory drive, use a Melling RV-type cam, burn a chip, and now you are set up with 275-300 easy and cheap HP, and its basically a bolt-in right down to the sensors and wiring harness. That way you're getting more oomph than a 5.3 without all the hassle. You won't win MPG contests, but you'll be surprised at how much it doesn't guzzle.
I used to build these for trucks and boats, except I tried to keep it cheap and I kept stock vortec heads and a Qjet or Holley. The one I built for my boat dynoed at 290 hp at 5300 and I forget the torque, but it was lots. I put one like this in a 79 Caprice (with a Qjet) with a TH350 and 2.73 gears and it could do 18-20 mpg on the highway. I know that holds no comparison to a truck with 3.73s towing a trailer, but I didn't think that was too bad.
