The voltage develops around loops.

Personally, I don't think it would work as you planned. The top and bottom of the air-core resonator are electrically the peak, and the node (zero-crossing), respectively of a quarter of the wavelength of the resonant frequency of the system. A wavelength of what, though? Well, if you answered "voltage" you wouldn't be far wrong. This means that the voltage at the top is at maximum, with respect to the bottom of the coil. Now you are putting a straight conductor between the two "poles" of the voltage (e-field) and the coil will likely short out the potential along this conductor. Tesla coilers learn fairly soon that you do not want to penetrate the cylindrical coil former with, say, loops of wire to lock it down, or terminators like screws through the sidewall, because if the coil is any good it will simply arc and spark away down inside the open tube of the former, between the wires or whatever you have stuck through the walls!

Another consideration is the inter-turn voltage. You have low voltage at the bottom of the first windings. Now you bring a wire with much higher voltage down and put it close together with the low-voltage windings. This works in a Tesla bifilar pancake primary because the overall turn count is low so that the actual voltage difference between adjacent turns isn't that great and insulation can withstand it. Tens of kV at most, in Tesla's applications. But if you try this with a big secondary you may be looking at _hundreds_ of kV between adjacent windings of the first and second set of wraps.  That is, if it doesn't just conduct down the middle of the tube before it can build up high voltage.

I  might be wrong but that's my first guess. It would be an interesting experiment to try, though. I've been working on a phase-locked loop circuit for resonating such things, but I don't have enough of the right kind of wire to try the bifilar secondary as you describe it. I'd want to use some heavier insulation, double-coated anyhow, than the magnet wire I have on hand. But who knows, I may "wind" up doing it anyhow.

(The project is coming along nicely:)
http://www.youtube.com/watch?v=XeQ5WnziKBA

Thanks, and certainly!
The resonator is about 420 turns of #27 single-insulated magnet wire wound on an oatmeal box and varnished with polyurethane spar varnish (soaking the cardboard of the box, so that it can't retain moisture). The primary is five turns of #12 solid copper house wire, closely coupled, wound on a bit of PVC tube that just fits over the secondary. The top capacity is two aluminum dogfood cans taped together with aluminum tape. The circuit is in three parts: a Phase Locked Loop oscillator made from a CD4046BE chip (under a dollar from Ebay resellers) and a few other components; a Mosfet Gate Driver  and Primary driver made with two transistors (2n7000 and 2n2222a) and an inverter gate from a 4049 hex inverter and the IRFP260 mosfet itself; and an Interruptor circuit made with a basic 555 timer with adjustable Hi and Lo timing. So the PLL circuit generates the resonant oscillation at about 736 kHz and locks, or tries to lock, to the resonance and will vary its frequency slightly to keep the coil in resonance as external conditions change. The Interruptor stage breaks up the PLL's output into tunable audio-frequency chunks with a 555 timer chip, which makes the "sputtering" noise (or a tone, etc) as the spark is turned on and off by the Interruptor. And the Gate Driver basically amplifies the current from the PLL chip so that the mosfet gate is charged up as fast as possible, and also shuts the mosfet off quickly. Interestingly, the Gate Driver isn't quite able to turn the IRFP260n mosfet on super-fast... so it actually reduces the "On" duty cycle at the Primary from the PLL's 50 percent, down to about 25 percent effective fully-on. This actually _improves_ the performance of the primary circuit because there is no point in leaving the mosfet on for longer than it takes to put full current through the primary. It's the turning off of the mosfet that produces the highest induced voltage and the Gate Driver circuit drains away the Gate charge very fast. The driver topology that I used is an inverting one, so I added another inverting stage using the 4049 chip so that the "on" state of the mosfet is also during the "hi" state of the PLL oscillator's output. This might not be strictly necessary but I think it helped with the phase-locking.