How to reduce noise causing by MOSFET driver?
1) Do not turn on one mosfet while the other isn't completely off. Not only the 250ohm resistor: add some dead time (time when no one of the mosfets is on) in your firmware, if you can;

2) Adding this dead time perhaps will obligate you to increase the output capacitors. And I agree with FakeMoustache that you need some bigger values of output capacitors than 100nF;

3) Very small value inductors, (again, as commented by FakeMoustache) more than once made miracles against my SMPSs output noises. You can begin with 1uH. If you want to test very high values of inductors, pay attention to the self resonant frequency (SRF) of it: I like to use at least 10 times the noise frequency;

4) Tip (if you aren't already doing it): be sure that you are measuring the noise very close to the output (and without the ground clip of the oscilloscope), in order to be closer of its true value and make good comparisons while testing;

5) Perhaps it won't change the output noise (but in the overall circuit), but I can't avoid my will to write you to add one inductor in the place of R99 and close to it one 100nF paralleled to a 10nF (attached to +10Vin and ground). This may attenuate the conducted noise to the +10Vin that you will experience.

Best regards