That is an interesting question about the border line case between needing a purpose made mosfet driver and not needing one.
The mosfet in this case is a logic level with Rds(on) of 32mOhms @ 2.5 volts and an input capacitance of around 1nF.
You are driving the gate with a ( push pull ) Arduino pin via a, say, 150R resistor and with PWM at 20kHz
My approach would not be very scientific. I’d do a simulation in LTSPICE to model the RC network at that frequency and visually estimate, from the wave form, the the % of the time the mosfet would be only partially conducting.
If that looks OK, and the device is otherwise within the specified parameters, I’d simply do a test to see if the mosfet gets hot under the anticipated load conditions.
I have seen some mosfet circuits where there is a (fast) diode across the gate resistor to speed the discharge of the gate, presumably on the basis that although an Arduino pin needs a current limiting resistor to charge the gate, it doesn’t need one to discharge it.
Incidentally, I found this post How to use a MOSFET as a switch. (controlled by arduino)where a 555 is used as a mosfet driver and discusses issues relevant here, with the main input from @terryking228 and @MarkT