Scott:
It would really help if you posted the results of the ping/arp scan of that network.
Results from the cmd prompt can be copied to the clipboard by right-clicking on the top of the window, selecting mark. Highligt and depress <return>.
If what you say is true about the hard-coding, it just reinforces "I hate Linksys" additude that I have. Their stuff doesn't work and their support is bad. My experience anyway. Once you make changes like that, the router typically has to re-boot or power-up/power down for those changes to take effect.
The new router needs an address on the network that doesn't conflict.
If the address is not on the original network, then the WAN of the new router gets connected to the LAN of the existing one. NAT and DHCP should be enabled.
If the address is on the existing network, then the LAN of the new to the LAN of the old should suffice. No firewall stuff happens. NAT and DHCP should not be enabled.
In both cases, we asre assuming that the router's address is not assigned via DHCP.