So the "Amp's remote" is just a logic signal input that tells the amp to turn on?
If the original "e-antenna" output from the head unit is intended to power an auto-extend/retract motorized antenna, then likely you could draw a couple of amps from it. If the motorized antenna input is like your amp (where the control input is just a logic signal), then it is possible that the "e-antenna" output from the head end can supply only a few tens of mA, maybe even not enough to power a relay coil directly.
Do you have an installation manual on the head end? It should specify what the "e-antenna" output is intended to drive.
Does the power amp already have an output that has 12v on it only when it is powered up? If not, can you bring a wire out of it that would?