I think getting enough inductance will be a problem. As said, the windings should be divided to reduce the voltage across across any layer to < 400V and a couple of layers of mylar tape wound over each layer or wound on a multisection bobbin with the same idea in mind. It is difficult to get the wire from one section where its on the outside back into the inside of the next section without an air gap and two insulating "washers" (suitablely sealed to bobbin).
LOPTs use a very thin(1/4" -> 1/2") and deep (> 1" depth) winding which was totally encapsulated( to keep the winding mechanically sound) to give them the >10KV voltage output, but it was still the least reliable part of an old CRT based TV/monitor. I suppose that you could get a bobbin made and as you wind, soak the windings in polyurathane varnish, which will help to exclude water, but I think you will still need some tape in to stop any turns being pulled down to a layer nearer the core, so at the worst case you have the inner turns at 4KV and a "pulled down" turn which should be on the outside at 300V.
Frank