Bill, I managed to shoehorn the timer circuit inside a box of the same size as your KillBill unit.
It's your lucky day because I happened to find a few mini relays which pull just a measly 22'ish millamps when energised. The whole caboodle pulls a total of 30mA, therefore it shouldn't put too much of a drain on your battery. You have the choice of using the normally-open or normally-closed contacts, rated at 1Amp max. - should be fine for your purposes.
Unfortunately, by the time I squeezed in the relay, and the screw terminals, the pot, the chip and all the other gubbins, means there was no room left on the board to have a protection diode for the power supply. You need to make sure to connect the red wire (plus symbol) to positive, else you'll end up frying the chip.
The on-board trimpot lets you adjust the on delay from 1 second to roughly 12 seconds.
I'll bung it in the post to you on the morrow.