Car bluetooth power delay- USB capacitor?
It looks like you bought a badly designed adapter, which is not tolerant to "brown-out" conditions. In cars, when starting an engine, the 12V outlet voltage drops or even toggles several times. All car-intended equipment are designed to withstand this power shortage.

However, adding a simple capacitor to USB cable might not be able to solve this problem. Since your power requirement is 0.3A at 5V, it means that the load is about 5/0.3=17 Ohms.
To hold the VBUS power for, say, 1 second, on 17 Ohm load, you will need more than 60mF (milliFarad!!!) of capacitance, based on crude estimation from time-constant RC = 1s. This should be a super-capacitor to do the job. 

ADDITION: You can try this 100mF super-cap/5.5V, it is just about $2.80. To avoid back discharge into the 12V - 5V converter, I would use a medium Schottky diode before the capacitor.