Do you have a wattage for the touchscreen display? They're quite power hungry aren't they? I'm guessing around 10-15watts.
I'm no expert though, so this really is just an idea that maybe you or somebody more experienced could take further...
Not sure what your vehicle is like, but mine has a feed that stays live all the time (one to the inside lights is like that for example). I think I'd use one of the always-live feeds as main power, controlled by a relay. I'd have a 555 timer, driving a transistor to switch the relay off 15 seconds after the ignition is turned off, and maybe chuck a big cap in there just to control any brief power dip during vehicle start.