Now we're talking....

Single 2kV secondary is nonsense, you won't find that really workable (>>2kV diode reverse voltages required), other issues will also arise (isolation, etc.). As small 1000-1500V medium-fast recovery diodes are still available, I'd suggest
a) Splitting the secondary into two (or better even three) separate ones, each with separate diode set and filtration capacitor, all stacked upon each other. But this may be more suitable for medium-high power application (couple tens of watts or more),
I'd suggest the more appealing easy solution:
b) use a multiple stage voltage multiplier. With a voltage trippler (which I think is pretty reasonable here), you would need just 3 diodes and 3 capacitors, all rated for 1000V, which is within reasonable amount (no voltage transients shall be present, due to the almost sinewave output of the resonant royer converter)
To get 2200V output, the resonant royer then needs to provide a sinewave with an amplitude of 2200/3 = about 730V (about 520Vrms).
Voltage tripler: referer 4example here:
https://www.electronics-tutorials.ws/blog/voltage-multiplier-circuit.htmlSelecting components for the voltage tripler should be straightforward.
The question remaining is the transformer. Me, as a guy working with power electronics quite often, I am used to design and prototype my own transformers. So I'd definitely go down the route to design and prototype one myself. If you are limited to off-the-shelf components, then well.. doh! Can't help much that way. You need to start looking, hard.
How one would design the resonant royer transformer?
You need to start with couple of things: Decide on the frequency of operation, output power and resonant tank Q.
The primary tank Q defines the amount of reactive power (VAr) in the primary LC tank just S = P*Q.
What amount of Q you may ask? That is a very good question, should be within a range of 2 to 10. I'd suggest staying on the low side (2 to 5 really). Otherwise you will have a rather large circulating currents in the primary, that will try hard to burn the shit out of your primary resonant capacitor. On the other hand, higher Q means more stable oscillation and nicer waveforms.
The other important thing to know is that on each transistor collector, the voltage has a shape half cycle half sine and half cycle zero. The amplitude of that sine is Pi-times the input supply voltage. So across the primary you will have 2Pi*Vin peak-peak, or Vin*Pi/sqrt(2) volts RMS. So now you should know also the primary voltage.
Now when you know the frequency, primary S (VAr, reactive power) and primary RMS voltage on the tank, you can calculate both L and C easily. (For example, reactive power on C is S = U
prim,rms2*2Pi*f*C, this is really basic stuff...)
So you have both resonant C and the required primary L. The L required will be quite small, so the transformer
MUST have an air gap in the core. (That is the difference between resonant royer, and the "proper" royer circuit that uses core saturation for commutation. Saturation - ie. has NO air gap!).
What you are left to do is a basic air-gapped transformer design with a required primary L and primary rms voltage. You need to optimize both the core size and air gap to get optimum use of the core. The design process is iterative, you need to pick a core based on experience and engineering estimation, calculate all the required windings including their current ratings and wire diameters, then obtain the overall amount of material in the bobbin to assess if it fits (does not fit = too small core) or if it will be left almost empty (= too large core).
I can not explain here how to design a transformer like that, this post is already too long for anyone to read.
//EDIT: As a sidenote, I have a feeling that most CCFL transformers are designed for resonant push-pull operation with the operational output voltage just around those 500Vrms, which would fit your application nicely!