good evening gentlemen
I tow a number of different trailers with a 2002 Avalanche and had the tow mode crap out on it, it turns out that a common problem is the switch wiring at the end of the gearshift base in the steering column breaks because they are the size of speaker wire. I took the column cover off and removed the gearshift, took it to the workbench and soldered the wires together and covered the repair with 2 layers of shrink wrap last summer and have had no problems since. It takes a little time but saves alot of money in a repair bill if it is indeed the problem. It looks like when pushing the tow/haul mode button nothing happens, if you wiggle the gearshift while pushing the button it will sometimes work, if it happens like this it's probably the wire at the base of the gearshift.
check out www.chevyavalanchefanclub.com
do a search on trans problems and the wire fix is there.
Hope you get it fixed