I just put HID on my '14 N1k and know a bit about electronics so I'll give it a shot.
The ballast is a high voltage, high frequency step-up transformer. The primary is driven by circuitry in the driver box and the secondary, as you noted comes out 23Kv. It's not just a transformer like your pole transformer or other transformer seen in most electronic circuitry. It's part of a tuned resonant circuit which allows it to very efficiently step the voltage up from 12V to >20KV. Anything that unbalances or "de-tunes" this circuit will cause it not to work at all.
Frankly, I'm a bit surprised your first setup worked. Did you use HV wire? Standard wire insulation is only rated to 600v and will arc with 23kv. Anyway, the added capacitance of a few feet of wiring may cause the circuit to fail. In my case, the ballasts are mounted to the front squirrel cage. I had one light that would intermittently not come on. I had neatly twisted the HV wire pairs together. Straightening them out fixed the issue. Mine are 35W Ledperf units and are wired directly to the original N1K headlight wiring. Since they draw way less than the original bulbs there was no need to add a relay or extra power source.
BTW, most likely, your old ballast transformers and drivers would be fine if you just dried it out. Let it sit a few days in a warm place or use a hair dryer. Also, shorting or opening a line should most likely not hurt the circuitry.
I know you said you wanted to mount the components under the seat but the critical lines are the HV lines from the ballast to the bulbs. These should be kept as short as possible. That said, did you heat shrink the individual wires separately? If you keep them individual and away from each other (not twisted together) they might be OK. Also, keep them away from the metal frame if possible. Running them inside the plastic fairing is much better than taping them to a frame member for some distance.
Keep us posted on what you find.