I have been researching this subject also and don't have a definitive feel for how well our TT batteries will charge yet. We just bought it new back in November, but have not been able to do a shake down trip yet due to the weather here in Oregon.
I do know my 5 ton, 12' dump trailer which runs on one 12v battery to raise the bed will recharge given an hour or two of driving time. That's after a substantial discharge to make a dump cycle. That said, it has blown the 40a fuse on my 2004 GMC Sierra 3500 before when leaving the trailer plug in while cycling the lift pump. This tells me there is somewhere near 40 amps going to the trailer battery. (Now I unplug before a lift and reconnect for the drive out.)
I read elsewhere that using tow/haul on the older (2000's) GM's will increase the charge to the trailer along with using the headlights to kick up the amps. I do use the headlights as our dump site is on a highway the requires their daylight use, but only Tow/haul when loaded. We have made over 70 trips and it always recharges for the next go-round.