There is no faster diode than schottky. The GaN itself behaves as a schottky with Vf ~ Vgs(th) + Vgs(off). Both are minority carrier devices, and suffer no recovery or time delay other than that given by their dimensions, RLC equivalent circuit, whatever.
These waveforms are on a time scale of single nanoseconds. It would seem the better question is: what diode can you even wire in parallel with the low side, that will provide the required properties -- namely, stray inductance in the diode-FET loop, and junction capacitance.
In other words, speed is limited by small-signal bandwidth, which is wholly dependent on layout geometry and switching impedance.
Have you considered:
- Using a GaN-specific controller?
- External gate driver?
- Those GaN-driver-and-inverter-in-a-chip parts?
You may not be completely screwed, with this combination (the controller plus GaN). A possibility is putting the series gate resistor in the negative (PGND?) path between controller and bridge, so that the negative swing can be clamped by a (smaller) diode, without upsetting the controller. Whether this is feasible given other limitations (e.g. permissible AGND/PGND voltage, whether the controller itself needs to be referenced to a global GND e.g. because of a programmable voltage setpoint or something, etc.), you'll have to figure out.
Tim