Sizing a powersupply for a gate driver
The PEAK driver current is 2A, not the continuous current. Your power supply needs to supply the continuous current to recharge the decoupling caps and the caps are what actually supplies the peak currents to the gate-source capacitance in the MOSFET since the power supply is too slow (or unavailable if it's a high-side bootstrap)

Also, it's not actually your gate driver peak current you need to be concerned about. As long as its high enough to turn your MOSFET on and off fast enough then what you really need to be worried about is your gate charge \$Q_{gs}\$ and switching \$f\$ frequency which determines the actual current your power supply needs to be capable of.

\$I_{average} = \frac{Q_{gs}}{T_{switch}} = Q_{gs}\times f\$