You could diode-block the supply feed and add enough supply
capacitance to ride out the sag, perhaps. Raw voltage would
still be over 12V running, plenty of headroom.
But maybe it's your car that wants fixed. Cranking voltage
should stay over 8-10V in good condition, but bad grounds,
weak battery, corrosion etc. all can detract.
If the load is modest, for this gizmo, you might even consider
using (4) NiMH batteries and designing yourself a charge
controller instead that only tries, when IGN is on. Which
might be as simple as a diode-blocked, resistor limited to
C/10 or so, feed. If C/10 exceeds your average current
draw then you're good.