Yes the tow package does have the hot lead in the center of the round factory tow plug, it will charge your battery, my '02 does.
You have the 42 RLE tranny in an 06. When has your tranny temp light come on? High altitude? Heavy Load? The guys in my club here in Colorado with that tranny have only had it come on when climbing a steep grade at a slow speed. Hauling a trailer over I70 with that tranny on the way to moab none have had it happen.
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