From the pictures, it appears that amp only turns on when power is connected to it and as you said does not have a remote turn on like normal amps. The best way to accomplish what you are trying to attempt with an amp that does not have a remote on is to use a standard bosch single pole double throw (SPDT) relay.
86- DLYON (+12v delay on from the power supply)
30- output to amp's +12v input
87a- not connected
87- Constant (+12v from battery)
The relay will complete the circuit and powers on the amplifier when the delay on from the power supply activates. And after the the power supply shuts off the remote line removes its power from the DLYON and the amp then turns off.