I might be way off track and I have no intent of testing this as it is high voltage and I don't want to dabble with HV yet... but...
From my understanding of it; would it be possible to produce the same effect of the Tesla's Hairpin circuit if you replaced the two capacitors with two bifilar Pancake coils wound in series? or is it not possible because the "plates" of the capacitance of the pancake are connected?

Hi,
I build a solid state hairpin circuit with a bifilar coil.
The coil does not replace the capacitors because, as you stated the plates are connected.

I use the bifilar coil, instead of the stoutbar. in between the capacitors. I don't want it to resonate, so I also added a diode in parallel to prevent this.

You can find it on my work bench if you are interested.
http://open-source-energy.org/?topic=3293.0