Are you saying that the router would be in charge of the IPs if it was between the modem and arduino?
Yes, on the router localnet only. Normally a router will perform a NAT to the WAN address on the router.
Can you access the router setup? If so, you may be able to assign the Arduino a "static" dynamic address. It isn't actually a static ip, but the dhcp server will reserve an ip and assign that same ip to that mac address every time it requests one. That has worked as well for me as a static assignment.