I don't like how many doublers there are... the LM3586 is a boost/flyback controller, with peak current mode control. It'll choke, driving much of a doubler.
Easy enough to change it to all-flyback, and also use fewer supplies, like the uh, U17 supply could be a zener shunt regulator from C55, and C54 and associated parts can go away. (If you have a PCB and kit, easy enough to DNP these parts, and strap on new parts, between pads wherever.) Would think POS5A and POS5GPS can be supplied from the input instead, whatever that is (7-12V I suppose). That leaves NEG5, POS250 + HV multiplier chain, Heater, and Blanking, which are all unavoidable. Of those, only the multiplier needs to draw forward-switched load current, which is annoying, but at least it's better than, well...all of them.
Shouldn't need many turns on a core that size, and Bmax = 0.1T or even higher should be reasonable. The big deal is: getting the air gap right for the required magnetizing inductance, and getting the windup right for the required (low enough) leakage inductance, for good efficiency.
Probably the primary can be a single layer per section (instead of two series sections as shown), and made in two or three sections, all of which get wired in parallel for the total winding. This is important for leakage. The other windings go inbetween these sections, with enough layers of tape to meet the insulation requirement. Probably, the HV winding should be by itself, then the little secondaries can be grouped together, and blanking maybe should be on its own as well just because of the isolation rating.
The blanking winding could also use triple-insulated wire, which can sometimes be salvaged from things, or maybe you can find some on eBay.
Tim