DC switchmode power supplies aren't designed to load share unless they are spec'd that way.
During engine cranking when the 12V sags likely can't be compensated by the supplies anyway, during input voltage sags while under load the input current would climb dramatically and could blow the supplies. Switching supplies work hardest under 'low line' conditions.
If it's a real bother, consider a small garden tractor battery feed by a 30 amp diode from the alternator or car battery, the idea is that the cranking voltage sag will be blocked by the diode and the small battery will keep the radio on with 12V.
A cheaper solution could be a 30 diode connected to 25V 1,000,000 microfarad capacitor that is connected to the radio 12V input.
The diode is 'forward biased' when the engine/ alternator is running supplying 13-14.5V charging voltage, during a dip the diode blocks the low voltage from dragging down the radio. Be sure to connect the diode the right way around.