The reference circuit looks incorrect to me. RSTn is active LOW, so I would expect the 10k resistor on the RSTn pin to be a pull-up and connected to the 3.3V line, not a pull-down to ground.
The W5500 datasheet says that RSTn should be held down for at least 500uS so I would expect a 5ms delay to more than suffice.
A GPIO pin on the 32u4 can sink a maximum of 40mA of current, but the current across a 10k resistor at 3.3V is just 330uA (0.33mA) so it should have no problem holding down RSTn.
A pause of 500us is generally indiscernible to a human so I suppose you could just tie the 32u4 RESET to the W5500 RSTn, assuming there is a convenient way to do so, and save a GPIO pin. A firm positive press will more than likely reset both, but I guess the GPIO pin approach does give you a guaranteed duration.