Automotive 3.3 volt supply that can survive engine start
New theory:

Keep in mind a significant voltage sag during engine start is normal.  For many years, automobile engines did not have electric starters because it was impossible. The starter motor would weigh 500 pounds.  The battery would weigh even more.  What changed is the realization that they were sizing based on the continuous rating, and they didn't need to. Electromagnetic equipment (transformers, motors) can be insanely overloaded for short time limits, because of the "thermal inertia" of the copper and iron.  Now the motor gets dramatically smaller and the battery is optimized for surge start. (at the expense of deep cycling). 

Series wound DC motors make torque proportional to current.  Voltage is irrelevant. Therefore the starter-battery system was optimized to deliver the necessary current even if it means significant voltage drop!  (and the ignition and fuel injection system must be ready for this.) 

So must you.  Prepare for the nominal 12V to drop to as low as 4 volts. 

I don't see that being a huge problem, as long as you are using a bucking DC-DC converter. It should cheerfully accept anything as low as setpoint voltage. 





First theory: (may help others with similar problem)

Because you have it wired into a circuit that is "Hot in run", like the fan blower.  

The system is being switched off by the ignition switch when you move it from "Run" to "Start".

Generally a key has 4 positions: 


Off - really off.  
Accy - turns on auxiliary loads such as car radio
Run - turns on all normal loads
Start - Only turns on loads essential to starting the engine, e.g. the ignition. 


Most circuits are "hot in Run" meaning they only are energized in Run mode, e.g. the fan blower.  They are shut off during engine starting to conserve power.
Some circuits are "hit in Accy and Run" such as the radio.  Again, off during engine starting.

So you are on one of those circuits.  You need to move your device to a circuit that is hot in Run and Start.  There are precious few of those - the ignition being one.  Beware: Line noise.