Hello Tobias,
Is your main concern the 360us delay before switching begins? I suspect that the delay is a combination of the time required to charge the VCC capacitor to the VCC UVLO threshold (7.5V typical) and the time required to charge the soft-start capacitor to a level greater than the COMP to PWM offset (1V typical). Check the waveforms on VCC and COMP to determine when those signals are at levels that will support switching on OUTA and OUTB