A simple, no teslaman, I do not know the wire gauge sizes recommended, nor the spool size you will need, would have done just fine.
Oh, plenty of people here know how to build your transformer, but they also know it will cost you more to DIY it than buy a used/surplus pole pig.
For example, this equation determines the number of turns vs. RMS voltage, core area in m
2, and peak flux density in T at 60Hz:
N = Vrms / (266.4 * Ae * B)
The larger the core area the fewer turns are needed to support the primary voltage at a given flux swing; something to keep in mind given the secondary will need something like 62x more turns (and will likely need to be wound in separate - or "Pi" - sections, thereby requiring even more room).
You want to use a C-core for a HV transformer, rather than E, because winding the primary and secondary on opposite sides of the C-core will provide much better voltage withstand capability vs. winding them concentrically or adjacently, as would be needed on the E-core; the increased leakage inductance of the C-core winding arrangement is also a plus here, as it will help limit short circuit current. I'd also say your estimate of needing 35# of laminations is about half off, though with a low duty cycle it might be okay. You will also likely have to wind the secondary in "Pi" fashion using quad-build #26 or #28 magnet wire (MWS is my preferred supplier for such), especially if you can't vacuum impregnate the windings.
So given all that a potential core would be one or two pairs of this one from Bridgeport Magnetics at a cost of $167 (not incl. shipping) per pair:
https://www.bridgeportmagnetics.com/shop/cd40x80x200-c-core/The core area of one pair is 3.15" x 1.64" = 5.166 sq. in. = 33.33cm
2 = 0.003333m
2So for a 240Vrms primary and 1.5T peak flux density you'll need 180 turns of wire, and to handle 33Arms the primary wire will need a cross-sectional area of around 7-9mm
2 (roughly equivalent to #8 AWG). The secondary will need ~62.5x more turns with a cross-sectional area of around 0.125mm
2 (equiv. to #26 AWG), and will likely need to be wound in five Pi sections of 2250 turns each, with either an appropriate air gap between each section or using a good insulating material (mica, PTFE, glass, etc.).
This is an awful lot of turns to handle on the secondary, and I didn't even check to see if the window area would be sufficient for the above winding specification, but I suspect you'll either need to double up on the cores to cut the number of turns in half, or else cut the cross-sectional area of the wire in half and accept very high I
2R losses (which would be okay for very low duty cycle operation - say, 30 seconds on followed by 3 minutes off).
You might still need to immerse the transformer in dielectric oil and vacuum purge any entrapped air bubbles both for cooling and voltage withstand reasons, otherwise you could go the "dry" route and vacuum impregnate with varnish/shellac (polyurethane can work for a one-off like this).
Etc. and so on... Hence why the general consensus has been to just buy the pole pig.
EDIT - correct duplicate sentence