ok, so not sure if this is excatly what you area already doing, just explained in a non-visual way....
have the screen powered by an IGN/ACC +12v
split the cam video signal
feed one of the cam signals into the lilliputs rearview video input
run a power wire to the camera from your reverse lights
also from the reverse lights, run a wire to the liliput +12v rearview input trigger
so far we have a camera that comes on with reverse
next run the other cam signal into your PC with whatever video capture card you are using
from the power supply of the PC, get +12v and run it to the camera (leaving the reverse light power connected to the camera still as well)
now we have a camera that still works on reverse no matter what, but also has power whenever the PC is on and you can manually switch to see it using your PC software
get another +12v feed from the PC power supply, and run it to a standard automotive relay (to the coil)
cut your original reverse light feed that is going to the lilliput rearview input trigger and splice in the normally-closed switch portion of the relay
now anytime the PC is on, the lilliput triggered input doesn't happen (but if PC is missing or off, the normal lilliput rearview input works with reverse still)
finally, run an additional feed from your reverse lights to trigger the joycon input that tells the PC to switch to the rearview cam screen
and so in the end, whenever the PC is on, the camera is on, and reverse triggers the PC to view the cam (or you can manually select it).
and whenever the PC is off or missing, the camera is only on with the reverse lights, and the monitor uses it's rearview video input.
the only problem i see with this simple solution is that if your startup controller starts booting the PC immediately, then you won't have rearview right when you start car because the PC is on, yet not ready to show video (bootup). this could be solved with an additional relay with delay circuit.
Running the Reverse +12V -AND- the PC power supplies +12V together is not going to work - you can't splice them together because it means when the reverse lights are on, the PC is going to be getting unfiltered power, and when the PC is on, so are the reverse lights. It is also potentially very dangerous, depending on what else is hooked up to that line. In my circuit I solve this with two normally-open relays that have the Inputs both spliced to always-on +12V, and the outputs are spliced. This prevents current from going the wrong way up the triggering wires. You could possibly use diodes to accomplish the same thing, but you have to keep in mind they end up dropping the voltage, which may screw up your ability to use a cheap linear regulator on the LCD (and on the reverse camera).
Using a Normally-Closed relay to skip the reverse-trigger when the PC is on is already in the circuit (The middle NC relay)
My circuit also has a delayed-on relay for the exact reason you specify. :)
I think the only other thing you mentioned was powering the screen via IGN/ACC - That was my original plan, and looks to be the typical setup, but I found out that if the PC is off, I get a bright blue "NO INPUT" screen on the LCD, so if the PC was off/disconnected I'd get that for the whole drive (except while in reverse) - so I've rigged it up to provide power to the LCD when the PC is on or if the reverse lights are on - both connected to two relays to act as OR logic.
FYI - I am moving forward with this design with some small modifications. First is that the timer circuit will be tossed out from the get go - I've got about $400 worth of Arduino parts and accessories on the way. This will also replace the Joycon EXR, too. I am hoping that after I get my system polished up I can sell ready-made boards to make life easier for everyone else. :)
The next modification is the addition of rotary switches on each output that let me set different modes of operation. Modes will be "Always On", "Always Off", and "Normal", with the LCD screen having an additional mode so that I can select whether it is on when the M4 power supply is on or when the PC is up and in Windows.