I'm a bit confused about what's going on in this thread, can I just check a few things?
You're switching your amp on with the 12V lead from your ATX power supply that's running off an inverter? And your Xenarc's also running off this 12V supply?
If you're getting 8V at the Xenarc's end, you might want to run a thicker cable from the ATX 12V line to your Xenarc because you're probably draining too much current for the cable that you're using.
You're fine to use the relay in this situation (try to buy a good quality 12V relay so you don't burn the coil out running it at specced latch voltage all the time). Check what voltage you're getting at the relay's connection, if it's too low it won't latch (as someone said before).
There's an error in your pin wiring, make sure that 86 goes to the ATX PSU's ground (the pins leading from the black wires on your standard ATX power connectors) and not to -12V. 12V to -12V is 24V, and you'll break the relay.