My 04 wont run with it disconnected.

 

I'd definitely double check all of the fuses, I'd think that circuit would be protected. I believe that the newer trucks have a sort of alternator duty cycle that is PCM controlled, could be something in that circuit or even the tuning. Also check the batt cables for corrosion.

 

Pulling a battery cable off while the truck is running used to work in the 60's, but I would strongly recommend against that practice with a computer controlled truck. You are liable to send a voltage spike through the system and damage a component far from the alternator. Have your battery load and alternator load tested. Good luck