The secondary would then be rectified and filtered then passed to a doubler to avoid insane turn ratio
You'd better put it though the doubler before you filter it

Is this for a one-off or do you have commercial (serial production) in mind?
For one-off, you can get plenty of variation of boards form china that have a SG3525 with step up.
And for the rest, there are plenty of commercially available transformers for high voltage applications. CRT transformers are obsolete, but there is stuff from electric fly swatters to cold cathode tube drivers and car bobines.
That old AN18 is a good start, for the rest 1kV was (is?) a pretty common standard for metrology stuff. Once I read an application note that put a lot of shunt regulators in series, and it really was a lot. I think that was for a 20kV reference, an the string of shunt regulators was wound in a spiral around a plastic pipe.
Do you want a fixed or variable voltage?
And at what sort of accuracy and stability?
The little gadget in the thread below goes up to 3.6kV.
https://www.eevblog.com/forum/beginners/how-dangerous-is-4000v/?all