I do not know what you mean by a flyback core. Perhaps it is the core from a line output transformer. If so, it will be made from ferrite and suitable for high frequencies. You are then feeding into a car ignition coil which is relatively a low frequency device. The ignition coil has a laminated iron core so will absorb a lot of power at high frequencies due to Eddy currents.
I suggest that you dispense with the car coil and use a Line OutPut Transformer complete with the high voltage output winding which will be suitable for up to 10kV or so.
The oscillator can also be run on this core, eliminating another souce of mismatch and power loss.
Invertors that I have made using a LOPT core, were wound with 1t/V bifilar wire and about 20kHz. I have not used the secondary, just wound my own to get 400V or so. The turns ratio will be the square root of the inductance ratio so you can find the turns of the secondary without unwinding it.