Having recently been experimenting with this on a PSC HVAC compressor, I can offer some input. First, if you are looking to optimize the power conversion efficiency, not only will you not change the capacitor size with frequency, but you will leave it in the circuit and power the motor with only 2 phases of your 3-phase output VFD. Obviously, this requires the ability to deactivate the phase loss detection fault of the drive.
As you reduce the frequency from baseline, as you'd expect, you'll see the auxiliary winding current drop. Due to the loss in torque generated by the auxiliary winding, you will see a bump in the current on the main winding most likely the result of a slight drop in RPM. For my motor, going from 60 to 50 Hz resulted in a 10% increase in main winding current with almost a 40% drop in auxiliary winding current. You can increase the capacitor size to increase the auxiliary winding current, but you don't see a corresponding drop in main winding current, I suspect because the resulting phase angle is incorrect. You just end up just putting more heat into the motor and drawing more power overall (and I'm talking watts, not VA), so it's really not worthwhile. Going the other way, you see the opposite trend.
If you kept the motor within roughly 10hz of its base frequency and or never loaded it near its full capacity (according to torque, not power), the change in winding current is not likely to be enough to matter. However, down to 20hz, you are likely to have some issue with heat generation in the main winding without some additional cooling. And I'm not referring to the effect of a slowly spinning cooling fan. At 20hz, based on my testing, at rated load, the main winding current would be well above its rating.
Your best bet would be to run the motor at base frequency and the highest load you can get, ideally its rated load according to its nameplate. Under these conditions, measure the current in each of the windings. Then, when you run the motor at different speeds, you can get an idea of what the relative currents are. If you enter operating windows wherein a winding's rated current based on your previous measurement is exceeded, you can either reduce the duty cycle or figure out a way to add some cooling. Keep in mind, any cooling you add must be in addition to any cooling the motor was expected to have at rated speed.
I am editing the above because it dawned on me that the test I originally ran was flawed and the discrepancy between the results and what I thought they should have been bothered me until I figured out what I did wrong. Basically, if you reduce your drive frequency, you'll want to increase the capacitance by the square of the difference in frequency assuming you are using a drive that maintains a linear V/Hz relationship. If you were operating over a small enough frequency range and needed maximum torque, you would size the capacitor for the highest frequency and hope you still had enough torque to operate at the lowest frequency. If the maximum rated torque of the motor isn't needed, the solution in the next paragraph is a viable option depending on the characteristics of the auxiliary winding.
Along the lines of not worrying about energy efficiency or requiring peak torque, it is also an option to remove the capacitor altogether and wire the motor to the VFD as though it were a 3-phase motor. As long as the motor doesn't have to start against a large load, it'll run just fine. I didn't document winding currents in this configuration, but I would expect the auxiliary winding current to be very low since the voltage across said winding when the capacitor is in play is well above line voltage. This would come with a corresponding bump in main winding current. It's also worth noting that depending on the electrical characteristics of the auxiliary winding (and in your case, the 2:1 resistance for the aux versus main windings is about the same as the motor I tested), the motor can be reversed via the VFD when wired in this configuration.