So I wound a pulse transformer for the lamp, but still can't get ignition below 300V.

Just to make sure, the larger electrode is positive, and the smaller sphere is negative?

The polarity should be stamped somewhere on the tube.
I remember something, then the cathode should be bigger, what is opposite then you just said, so better look for some documentation and/or marking.

There is something like a "+" mark on the tube, but I'm not 100% sure if that's the polarity indicator or just a part of the marking.
I just tried reversing the polarity and it doesn't ignite at all, so I don't know 

Could this be because of the age of the lamp? maybe air has diffused into it, making the ignition harder?

Maybe.

The bigger electrode will be positive, as it has to withstand a very high temperature from electron bombardment.

The wire from the pulse transformer should be both short and well away from any surrounding metal, to reduce capacitance. Preferably it should be PTFE insulated as well, though just using HT solid core non resistor wire works as well. The type used for the final anode in a old TV or CRT monitor is perfect. You might try using a higher voltage supply to drive just the trigger supply, or try another non wound capacitor  to provide the energy to the trigger. Your 1.5uf unit can be made up using 3 0.47uf 630V mylar units in parallel, as this will reduce the inductance and foil resistance over a cylindrical unit. Preferably do not use a class X or Y unit, as they tend to have very much higher internal resistance films. Use the brown capacitors from old monitors that are in the area around the LOPT as they will have better pulse rating.