They say one picture is worth a thousand words so two are probably even better. I stayed up till 3:00AM doing these so do look at them.
They show the relationship between the current in the motor windings and the torque being produced in both single phase and three phase motors. I do not claim absolute accuracy here, just an attempt to show the general relationships. The point is, the torque in a single phase motor goes to essentially zero two times in every cycle of the AC current in it's windings.
This is essentially a 100% variation in torque that occurs 120 times per second. I can not offer direct proof that this will be transmitted by a belt drive to the spindle, but it would certainly be transmitted via a gear drive.
On the other hand, because a three phase motor uses three phases that are out of phase with each other and all three of these phases contribute to the total torque, the sum of the three is always a positive number, never zero. In fact, the variation in torque is closer to 12 to 15 percent. Far, far lower than the 100 percent variation of the single phase motor.
The mass of the rotationg load (spindle, chuck, and work piece) will certainly have a flywheel effect, but this will be the case in both single and three phase motors. So the difference in the motor's torque will still be a factor.
My drawings show simplified motors and as the notes on them state, real world motors generally have more poles on the rotor or stator or on both. But this does not in any manner negate the conclusion. More poles would simply produce motor that rotate slower. The sequence between adjacent poles would be essentially the same and the resulting torque would vary in much the same manner. The motor's sped will drop more due to the load when the torque drops to zero than when it only drops by 12 to 15 percent.
As for the vibration of the motor being transfered to the lathe via the mounts, that could also be a factor. However, a three phase motor can vibrate just as much as a single phase one and with a well balanced motor it may not be as much of a factor as the varying torque. I suspect that the variation in the torque is a much larger factor in this situation.
Make it fit.
You can't win and there is a penalty for trying!