I have never used the lighter socket cord, so this is just a wild guess, but the voltage would drop with more current flowing. resistance anywhere between the master buss and the cord will increase the current flow, possibly the iFly is more voltage sensitive than the Garmin unit is. I'd check and clean every connection from the back of the lighter socket to the circuit breaker, it the breaker is shared with other circuits. Both sides of the breaker if it's dedicated.
I could be way off base also though . . .