To be clear, I don't know that

an instantaneous switch/relay that does what a press of the power button would do

is something that you can buy off the shelf as a packaged component. You might have to build a small circuit to do it. You'd want it to wait for power to go from 12V->0V and have it trigger a normally-off switch to on for a half second or so.

I'm not an electronics engineer by any stretch of the imagination, but this seems like something a 555 timer IC would do.