I'm going to mount an LtAP in my car connected to a house battery and a 12-24V DC-DC step-up converter.

I'd like to make an automatic shutdown script for the LtAP that will shut it down after a certain number of minutes without the device talking on the LAN (or some other criteria to shut it down when it hasn't been used in a while).

I'd also like to find a way to power the device back up without having to manually re-apply power to it. From what I can tell, the only way to power it back on once I've issued the Shutdown" command is to actually disconnect it from power, completely, then reconnect a power source to it.

I'd also like to find a way to power the device back up without having to manually re-apply power to it. From what I can tell, the only way to power it back on once I've issued the Shutdown" command is to actually disconnect it from power, completely, then reconnect a power source to it.

Correct.
I'm not sure Wake On Lan is supported by MT devices.

Wouldn't switching the key in the car perform that power off/on cycle ?
And if you keep it running as long as the engine is running, it wouldn't have to be shut down in between ?
Just thinking ...

I'm not sure Wake On Lan is supported by MT devices.

it does

/ tool wol interface=ether-x mac=00:01

I'm not sure Wake On Lan is supported by MT devices.

it does

/ tool wol interface=ether-x mac=00:01

That's for SENDING WOL packages to another device.
https://help.mikrotik.com/docs/display/ROS/Wake+on+LAN

And if you keep it running as long as the engine is running, it wouldn't have to be shut down in between ?
Just thinking ...

I don't see how... If I want to avoid running the car's starting battery down, I'm going to have to have to run the router off a house battery. That being the case, starting the car won't re-start the Mikrotik...

Not sure how it goes with your car but if I switch off the key/engine, all power goes down after max 15 minutes.
Unless I press a button on the screen to continue.

Since the OP is talking about the "house" battery, I am assuming he is talking about something like a recreational vehicle - not a "normal" passenger car. If that is the case, the house battery stays hot all the time.