So you have an adjustable electronic delay timer that draws only a very small mount of current when it is working and you have powered it from one the wires on the light switch.
... Observation ...
The fact that this is an electronic delay timing circut that is only drawing a little current when it is working means that it will be very susceptible to electrical noise in the power circuit path. Without getting out my BJ8 wiring diagram, I think the voltage regulator is in the path to the power of your new toy. If I had to guess what was going on without asking you to check some things, I would suspect that the electrical noise generated by your voltage regulator is causing the circut to be reset/triggered by noise. When you turn the lights on and off, that can also generate a noise spike.
There are some things that you can do to eliminate various parts of the system to see if the " bad behavour " goes away .
I'll just list them out as they come to mind. Before I do that, we should get the exact symptoms straight,( yes or no );
s1.. When the motor is off, you don't have any problems even if you operate the light switches ?
s2.. With the motor running, you don't have any problems UNLESS you operate the light switch ?
Things to try one at a time. Do your best to see if it fails after each step;
1. Disconnect the wires from both door switches and wrap the exposed wire with tape to prevent an accidental grounding. Test it. If it still fails, reconnect these wires and go to the next step. If it stops failing, one of the door switches is defective or not adjusted right.
2. Obtain a small 12 volt battery and disconnect the power and ground wires for the circuit from your car and attach to the small battery. Now your circuit board's power is isolated away from the car's electrical system.
Start your car and test. If it still fails then the circuit board has a problem but only when vibrated by the running engine. I suspect that it will work without any problems when attached to the separate 12 volt battery. If it does, then this proves that the source of your problem is electrical noise. If you confirm this, let me know and then we'll go through some things to see if we can get it to work .
Ed