Connecting a low-side gate driver to a FET with current sense resistor
Because it is the capacitor between the gate and source that I am
  charging/discharging with the driver, I am inclined to connect the
  gate driver's source, as this way the sense resistor will not degrade
  di/dt into the gate.


If the current sense resistor is small enough it won't have a significant effect. And it should be small, to reduce conduction losses. The changing Gate charge at turn-on will cause a spike in the sensed current, but this can easily be ignored. In most cases the Gate drive will already have some resistance (to limit Gate current and prevent ringing) which will probably have more effect on di/dt into the gate.  


  However, if I connect the gate driver's ground to high side of the
  sense resistor, that means that the reference for the input to the
  driver will move with source current.


That's not good. Many Gate drivers have a logic low input voltage of only 0.8V, which could be compromised by voltage drop across the sense resistor. 

One way around this is to 'float' the entire control circuit, or (essentially the same) monitor current in the negative supply lead. This works when the power source is a battery or dedicated power supply, provided controller current is insignificant compared to motor current (or can be subtracted from the reading). However it may not be practicable if the negative supply ground is shared with other equipment. It also distorts the current waveform as the bulk capacitors smooth out supply current, so it might not be suitable for precise 'cycle by cycle' peak current limiting.