Two problems:
1. What signal level did you measure inductance with? Guessing it was small, a few volts, not the full 240?
This matters because the high-permeability laminated iron core is nonlinear. At low field, mu is relatively low (initial permeability). At higher levels (say, 10s of V at 50Hz) it begins to rise up to its nominal value (which may be a tenfold or greater change). At still higher levels (>240V at 50Hz), it falls again, rather sharply this time, as the field "saturates".
2. There's not much you can do on the secondary, well, besides putting an inductor in parallel. But if you had an inductor of the right size, you'd just use a slightly different one instead of the primary altogether.
But there is still something you can do: take the frame off, and with the help of a utility blade, flat screwdriver, hammer, and heat if necessary: remove the laminations from the bobbin. Restack them so all the 'E's are on one side and the 'I's on the other. Put the frame back on, cinch it up, and measure again.
What this does, is remove the overlap between laminations -- instead, field has to flow entirely through each piece, then jump the small airgap that is inevitably present between the E and I stacks. You can control this by adding spacer material (preferably paper or PET film) between them.
Air gap acts to reduce the effective permeability of the core. It also acts to stabilize it. While the core material itself might vary over a factor of 10, if the air gap is ten times more significant, the core variation can only contribute a 10% error, i.e. going from small to large signal only increases the inductance by 10%.
For small air gaps (gap length much less than the width/height of the core face), the air gap is inversely proportional to the inductance. So you shouldn't need too many tries to home in on the right value.
And the final part: the core still saturates above 240V. Air gap has little to no effect on saturation voltage and frequency. This solution is not suitable for a resonant ignition method. If you have a one of those bimetallic starters, or an electronic one, or the tube just does it by itself, no worry.
Oh, a final final part as well... is your transformer rated for the desired 100mA primary current in the first place? In other words is it 24VA -- roughly the size of a fist? If smaller, it'll run rather hot, or burn up completely. (The pictured unit, with apparently (2 * 6V) * 3A = 36VA, would do fine. I guess you've picked that picture out of a catalog, check its dimensions for comparison.
)
Tim