The circuit appears to be a flyback converter. MB3800 is a switching controller:
http://www.icbase.com/File/PDF/FJS/FJS00590106.pdfThere appears to be a MOSFET driving the primary, which is obviously the primary from the connections and the multi-strand (litz) wire used. The secondary is fine wire, probably around 10-40 times the primary turns, for the 80V output. VFDs need modest voltages at low currents (a few mA?).
The other secondary (three middle pins on the primary side) must be filament. VFDs are actually very small vacuum tubes; a microscopic filament spans across the front, emitting electrons over the electrodes. Oh, and this winding connects to the midpoint of the HV secondary, they must need some negative cathode bias -- hence the two diodes on the HV side. That explains that.
The transformer is probably massively oversized for what it needs to do (ER11 can probably do 5W or more, in an optimized design; this is probably less than 1W?), which may mean extra cost, but at least means few turns are needed, saving wire length and core gap.
A flyback requires a gapped ferrite core to operate (or other core types), so there should be some gap. I don't think the core is so big they could get away with none. But it may indeed be small, like 0.1mm.
Shimming the core is an acceptable substitute, but mind that the air gap counts twice: once for the middle peg (normally only this is ground down), once for the outer limbs. If that means 0.05mm spacing, well, maybe you can find some plastic film or tape to get there? A micrometer will be helpful.
A note, use insulators only. Any conductor in the gap will be blasted by the field. So like... just in case you had the clever idea to use the feeler gauges that way or something.
Tim