Personally I would not use an FFT for fine measurements.
A better method would be shift the 1kHz down to DC and then plot the phase as a function of time.
The width of the FFT bins will mate it appear like it is one frequency, but it will be a chirp in frequency,. And with any spindle hole offset, it will be a chirp with with a sine wave riding on top of it.
There will no now way to get the sample rate high enough to have the FFT size be high enough to get any sub Hz resolution.
And with the chirp and sinewave it will be smeared within the bin to all buggery.