Finally, the higher the high turns ratio (in either direction) the higher the leakage inductance, usually, (and the higher the losses from proximity effect, which, fortunately, does not apply to the flyback).
Hm? Explain?
Proximity effect occurs in any multilayer section. It's simply the image current of one layer induced upon the adjacent layer, that happens to already be carrying the same current in the same direction.
It probably isn't important at these currents and frequencies -- the wire used will be quite small.
It's like how the smallest wire I have in stock is 37 AWG, so when I want to wind something for high voltage, well, it's automatically capable of ~30mA continuously, even if I only needed a few mA...

So, which, actually... thanks, I never thought about this before but it's quite true:
In any flyback transformer, with multiple layers of windings,
regardless of interleave, proximity effect occurs.
This is simply because the primary and secondary currents are not coincident, so they can't cancel out their proximity. (Whereas you can do this in a forward converter with interleaved single layer sections.)
So flybacks with many layers are somewhat prohibitive, and one or few layers are preferred, which means wider bobbins should also be preferred (i.e., to maximize turns/layer). Cool!
(Interleave is still critical for leakage, however!)
While the clamp diodes in the two-switch variant do return all the energy stored in the leakage inductance back to the supply, while they are doing so the output diodes can't deliver energy to the load... So, a lower turns ratio secondary feeding a half-wave doubler/tripler/quadrupler could be highly advantageous here for that reason, as well.
In any case, the 2-switch discussion is spot on. Also, the secondary peak voltage rating is the same (total) whether the secondary is whole, or broken into sections with a diode each.
Incidentally, HV diodes are all just stacks of lower voltage diodes -- I don't think they can make high speed (<= 250ns or so?) diodes over 1.5kV or so. (In fact, an industrial "hockey puck" rated for 6kV is "high speed" if it has a recovery of 2us!) So you see V_F climb -- multiple junctions in series.
The trick is, they're matched diodes, and packaged together, so recovery and capacitance are equal. This is a lot harder to pull off with discrete diodes*, so there is value in the high voltage types.
*What happens is, one recovers first, and gets all the voltage drop and immediately goes into avalanche, burning a huge amount of power. Then the next one recovers, the total voltage drop rises and not as much power is burned, and so on until the total avalanche rating exceeds the applied (reverse) voltage, and conduction stops. And it's a runaway condition, because recovery gets slower at higher temperature.
At least it's not as brutal as with a forward converter, where the reverse current is switched in -- a flyback in DCM has soft recovery, with dI/dt set by the transformer inductance.
Actually, an even more subtle point is that recovery itself is a sort of dynamic avalanche condition: effectively, the off-state (blocking) voltage of the diode increases over time, eventually reaching its nominal rating. This probably happens simultaneously for all diodes in the stack, but the result is still that one diode dissipates much more recovery loss than all the others.
But anyway -- if 20kV diodes are unobtanium, at least the secondary can be split into sections, and as many 3 to 10kV diodes can be used, which are hopefully more obtanium!
And if you put all of these conflicting issues together you start to see why I keep recommending a flyback with a voltage multiplier,
Also -- note that a multiplier needs a different control circuit than a bog-standard (current mode) flyback. The multiplier draws current during switch-on, so either a current-limiting switch is needed, or something that acts in the same way. It may be adequate, for example, to use a normal peak-current-mode controller, but increase the transformer's leakage, so that the switch turns on for a moment, charges the multiplier through the leakage inductance, then turns off. (The increased leakage would suck for a one-switch converter, but would be very much in favor of the two-switch or full-bridge version.) The operational result will be, lower maximum power output at low output voltages, because the switch turns off much sooner than it would otherwise. If you don't need a wide output voltage range, and don't need a lot of startup current, that should be quite feasible.
A current limiting switch (i.e., a linear device) is probably completely out because of heat and size.
Proof that it can be done, though:
https://www.seventransistorlabs.com/Images/Deadbug_Sch.png This is hardly a watt, and the switching transistor can get quite toasty under some conditions, so it's hard to recommend at any kind of scale!
Yes, a properly designed (and protected) LCC or other resonant converter would achieve a higher efficiency and likely take up less space, but forget this being the deep end of the pool; at that point you are swimming with the sharks out in open blue water. If you have sufficient dedication and budget (in both time and money) to dick around with this, then by all means attempt a resonant converter design - it will be better for the job - but if the real world has placed practical restrictions on you then, no, none of the exotic topologies are really appropriate. Note that I am not saying you can't or won't be able to get a more exotic topology working - who am I to judge what you are capable of? - I'm just saying that it will be a much more difficult task, relatively speaking.
And speaking of budget -- do consider hiring a consultant to help guide you. Or, heck, just contract out the whole thing, if it just needs to be done -- give or take how many lessons you want to learn from the project as well.
Which, on that note, I'm pretty busy right now actually, but if your timeline extends to late summer I'd be glad to help. Otherwise, there may be high voltage / switching professionals here who are available; doesn't hurt to ask.

And you won't be running nearly as high as a switching frequency as you appear to think is possible - after all, a mere 10pF of stray capacitance has an impedance of 160k at 100kHz, so a 10kV square wave applied across this will result in 63mA of current. Whether that current is flowing across the junction of a diode that is supposed to be off, or across a transformer secondary, it amounts to a pure loss...
Well..... that's the whole point of the resonant converter, right?
It does tend to be loss, in a traditional flyback converter that needs wide bandwidth and can only discard energy that gets trapped in reactances.
It does seem unlikely to construct a transformer with bandwidth of multiple MHz (that would make a 20-100kHz flyback feasible), so that is a point strongly in favor of resonant.
The tradeoff is then, a flyback that's easy to design but has terrible efficiency, or a resonant that's hard to design but easily gets good efficiency.

Tim