How would it do with the stock tranny that I have for daily driving? Only time I tow would be a small trailer or a boat. Not too often. I already only get 11MPG anyways.
That all depends on your control of your right foot. The trans won't know what's in front of it- 5.3, 6.0, 7.0... It only feels how much power/torque is put on it. You could put a 1,000 HP big block to it and it'll live just as long as it would with a 4.8 hitting on 6 cylinders as long as you don't apply any more than about 1/4 to 1/3 of that 1,000 HP.
That being said, there's no point in swapping in a more powerful motor if you're not gonna use it. The trucks that got the 6.0 got heavier-duty transmissions so that they'd have the same general life expectancy of the weaker motors with the weaker transmissions.
Decide what you ultimately want and will have for a motor and buy a trans or build up your trans accordingly.
If the current trans is fine and you end up with a stronger motor, just know what you have and drive it accordingly. It may live plenty long enough for you or it my go all Tonyrodz and crumble under pressure.
If I were in your shoes, I wouldn't let the trans hold me back from a 6.0 swap. Like what has been said, drive it accordingly (minimal asshattery on the street, no full-throttling when pulling heavy loads, etc.) You're gonna burn the tires at least once. This doesn't put nearly as much strain on the trans as hard towing does because the power is transmitted through (and out) of the tires.