Voltage drops when turn on electric fan

I dont see how! Either a Alt problem or a Regulator problem.

Or a wire capability problem that adding another small load from a relay to the system will make slightly worse. :D
He said he's running the supply right off the battery post, so you can't get much more direct than that so adding a relay won't make it any better.
It will just add the small load of the relay on top of the current problem.