Martin,
Just a thought. When you start it on the lathe in full working mode, you are adding load to the motor. This, in turn, will make it more difficult for the motor to run up to speed and this will cause the motor to take more current. I don't know this lathe, so this question may be somewhat pointless, but by any chance does the lathe have a clutch? If so, is it fully disengaged when you start the motor as, just as in a car, if it is engaged, then this will increase the load on the motor.
Also, some people who have a lever operated belt tensioner, have been known to use that as a pseudo-clutch.
Finally, a starter won't stop any overload from being passed onto the mains supply. Well, it will, actually, by virtue of causing the starter to trip out and disconnect the power to the motor, but then you could be in an even worse situation where the motor never gets up to speed.
It is my understanding that the purpose of a standard starter incorporating a no-volt release and thermal overload is that the thermal overload is to protect the motor against fire or burnout in the situation where it is stalled. The thermal overload has a short delay in it (bi-metalic strip overwound by a small heating coil) which should be sufficient to allow the high initial starting current to diminish to normal levels without tripping. What this means is that if the thermal overload is sized correctly, it WILL NOT stop the high starting current from being passed back to the mains supply.
I fear that a starter will not cure your problem. Unless, and this something I know very little about, it incorporates some sort of slow or soft start by limiting the amount of initial starting current. Suffice to say that my lathe starter has no such device.
Regards,
Peter G. Shaw