I assume you have connected one of the CL ports to the WAN input on the Linksys. You NEED to do it this way.
Without you doing something to change things the DHCP server on the CL will assign an IP address to the Linksys connection. You can not allow that assignment to happen because it may assign a different address each time it connects.
First find out where the CL starts giving out IP addresses. This is typically 192.168,n.m where "n" is zero or one, and "m" is 1 or 101. The pick a high number for "m", nut less than 255. "n" needs to be the same. For example 192.168.0.161.
Then go into the Linksys settings and force the WAN address to the number you have picked. The mask should be 255.255.255.0. If it requires a gateway or server address, use the same IP string as above, only with a 1 as the last number.
Then go the CL router and set the DMZ to the IP you have chosen. Save all settings and power cycle both routers.