Need Capacitor for 12 volt Circuit
Either try a big capacitor across the RF receiver being fed through a big resistor which will more or less isolate it from the sudden transients when the circuit is triggered, or (the more sophisticated, but more reliable solution), use a boost converter to feed the RF section so that the battery voltage is irrelevant, the boost converter will always supply a voltage above the battery voltage (and it'll be a fixed, regulated voltage too) so that the RF section gets a steady supply voltage under all conditions.