Mmm, not a bad idea. I think a better idea would be a large capacitor charging from the LED output with a variable limiting resistor, and a very high resistance bleed resistor to discharge it when the LEDs are off.
Then, the LEDs would charge the capacitor until it reached a high enough voltage to trigger a transistor which drives the shutoff relay.
Trim the limiting resistor so that the short ~1 s flash from a hit doesn't charge it very much (and the bleed resistor will discharge the cap so it doesn't build up from multiple hits), but that the longer ~15s lit period when dead *does* trigger it. This way it would be independent of the number of hits required, and the period between the hits shouldn't matter. (If you tried to accumulate hits, the internal leakage of the cap might allow it to discharge between spaced out hits).

It still *may* be finicky, as heat will change the resistance of the limit and bleed resistor. Might also need to use good tolerance components if you wanted to build more than one, though the adjustable limiting resistor should help tune it.

Worth a shot I think. But my analog design sucks...so I'm not sure of the component values to start with.

R1 is the adjustable charge timer.
R2 is the bleed resistor. Would need to be big enough that the cap can charge within a reasonable period of time, but small enough that it discharges reasonably quickly.
Note that the LED+ is directly connected, and LED- is switched on the DBC. It might be easier to connect to the PIC chip before it drives the BS170 fet that drives the LEDs, as that pin is driven positive.

D.