Will the large gate current when turning on/off power mosfets with gate drivers cause problem to limited current power supply?
800mA peak current for a short amount of time will not damage the LDO.  The gate driver should be accompanied by a capacitor on the Vdd pin.  The capacitor will supply the peak current and not the LDO.  A 10uF capacitor is much larger than the gate capacitance.

The LDO has to provide the net (average) current for the gate driver.