Hi,
A gate driver delivering too low voltage is simply "unsuitable" independent of it's current capabilities.
Choose another, suitable driver.
As the MT3608 is non isolating, you can just use it for a low side MOSFET.
The use of it makes no sense in my eyes.
It's input is DC, it's output us DC and both are connected via GND...thus (the DCDC part) it should not be included in the gate loop.
(Only the output capacitor)
RMS is not useful here, better use "average" ... and average current should be rather low. It should be equal to
Q_gate_tot x frequency.
I assume any electrolytics - with it's high ESR - is not suitable to support the 22uF MLCC.
Any additional MLCC will not bring much benefit ... because of the low Q that is drawn by every switch ON.
Klaus