The DSATX has an additional 12v output.
I'm not certain if that is controlled by the shutdown controller timings or not. If not, you can use that 12v output for your Lilliput, and it will do exactly what you need.
Another way of accomplishing this is to use the egg-shaped regulator to power the screen directly form an ACC-controlled line. That is probably a far simpler means of accomplishing what you want.