On the 48RE
2003-2004.5: The shifter stalk button locks the transmission out of overdrive. When pressed, the trans will lock the torque convertor in 3rd (1:1).
2005: The button is a tow/haul button. It does not lock the trans out of overdrive. It does allow torque convertor lock up in 3rd, and lower speed lock up in 4th. Does transmission downshifting while slowing down. Manually selecting 2nd gear will lock the torque convertor above 25mph.
2006+: Press button once to engage T/H (acts same as 2005 model). Press button a second time to lock out OD.
Well, since I've got the 05 (and I have not and will not get the computer flash to make my button a lock out only button), I tow in overdrive nearly all the time. My truck climbs all hills at 65 with my sig trailer (~7k with street bike). On heavier trailers I will 'double-tap' to force a downshift to 3rd and let it pull the big hills in 3rd at around 60.
So OP, you can safely run in overdrive. The only issue you may have is low boost/higher EGTs with the slower spinning engine.