As Wilt indicated, do not connect this setup via the cigarette lighter circuit (as shown in you image above).
The flash will just require too much amperage for the lighter circuit.
Connect it directly across the battery terminals and use some sort of fuse. Position the fuse very near one of the battery terminals, not way down the line in the wiring.
And you will need a "true sine wave" converter of sufficient capacity, not just any generic converter.
You cannot just run into any K-Mart, grab anything they have, and get good results. This is going to be something a bit more than plug and play.
More like, think a lot, carefully design your setup, choose gear carefully, modify gear and car to conform to you needs, then play.
There are several threads here on homebrew inverters to power flash units using stand alone sealed lead acid batteries (SLA). They would work exactly as your intended use, but with the substitution of you car battery in place of the external sealed lead acid battery.
Enjoy! Lon