I can't help with modifying the spindle but can suggest buying a 3ph speed controller. These are now easily available, but most we see advertised to model-engineers are designed primarily to run 3ph motors from 1ph domestic supplies. Nevertheless, if you have a 3ph mains anyway, I'd have thought the same suppliers can provide a speed-control alone. Worth enquiring.

However… the motor still needs to run fast, at or fairly near its normal maximum. My inverters have their "red zones" at the low, not high, end of the scale. I am not certain why. It might only be for cooling and one or two people have reported replacing or augmenting the motor's own fan with a separately-driven one.

I am guessing though that if this lathe is actually a special type of glass-cutting grinder, it needs high spindle speeds?

For that reason, it would be best and anyway is by far the easiest solution mechanically, to keep the belt drive even if you leave it almost always on one, optimum setting. Changing the electrics is much simpler! Changing a belt drive can involve a lot of machining.

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Regarding machining the spindle, that is best done on the lathe itself, if only a simple diameter reduction, if the lathe has conventional screw-controlled slides. It may mean fitting a temporary tool-post. If it needs a nose-thread re-cutting and this lathe has no facilities for that, you would have no alternative to having it machined on someone else's lathe.

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As your enquiry suggests this is a highly-specialised machine-tool, it might help if you can supply photos (I think you have to create an album).