Ignition coil arc initiation with protected low voltage continuation?
Here's my suggestion for some experimental work.

Since you are looking for a DC discharge maybe all you need is a single high voltage pulse of quite short duration to trigger the arc. So, maybe try a camera flash circuit put through a small step up transformer to initiate it. To protect the supercap a couple of high value inductors and behind them some zener diodes to limit any voltage higher than the capacitor charging voltage.

You will likely need a scope that can handle high voltages to see exactly what is happening - there could be some weird resonances involved. Also, initially, leave out the supercap and just see if you can trigger a one-shot arc successfully. If you want to go beyond that, google "stun gun circuit" for your high voltage generator of more simply try something like this from eBay Ignore the hype - you will get more like 15-50kV from it - the wider the discharge electrodes the higher the voltage and the slower the repetition rate