Some clever networking devices, when sense that there’s something in their network port jack, start process of 'blocking, listening, or learning states", or even line training to identify what type of cable is connected. This process may take up to a minute, thus device plugged into router will not be able to access the network for quite a long time after it was plugged OR after it performed link down/up operation (note: W5500 performs link down/up when hardware or software reset).
Check this article from Cisco. Cisco devices are having PortFast feature:
If you connect a workstation or a server with a single NIC card or an IP phone to a switch port, the connection cannot create a physical loop. These connections are considered leaf nodes. There is no reason to make the workstation wait 30 seconds while the switch checks for loops if the workstation cannot cause a loop. Cisco added the PortFast or fast-start feature. With this feature, the STP for this port assumes that the port is not part of a loop and immediately moves to the forwarding state and does not go through the blocking, listening, or learning states. This command does not turn STP off. This command makes STP skip a few initial steps (unnecessary steps, in this circumstance) on the selected port.
which allows configuring port not performing start-up operations when connected, and go directly into forwarding operation.
If your router does not have such option, the only way I see is connecting W5500 to a simple hub device, and then connect that hub device to the router of the network. In this situation if W5500 will perform up/down of its PHY, another hub’s port will still stay up for the router device.