Ok, I think we are going to misuse this relay thingy, probably at least using not the way designer intended. But should be still safe.
As in the picture, connecting the switched negative through a resistor seems to do the trick. The timer starts/restarts every time when the line connects to the Batt neg (on the leading edge only). So if switched negative stays “active” the timer will stay on only set time. A “pulse” is needed to restart the timer - if pulse comes before the expiry the timer starts over again .
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As we are misusing the circuit, the resistor is needed to limit current into circuit and the chip below the board. Without the resistor the circuit draws more than 30 mA into the chip itself. I’m afraid that would pretty soon fry the chip. It seems that the resistor could be soldered actually to the board - by replacing a one there.
I did some experimenting and at 13V it seemed to work reliably up to 47 kohm. Didn’t yet check if/how undervoltage would change the threshold. So I think anything 5-22 kohm would be fine not frying the chip and giving decent marginal against voltage fluctuations.
I'll try to verify next week if there are any revelations...