Linux – When interface receives an IP address, run a shell script

Add the line iface eth0 inet manual to the /etc/network/interfaces file. This will tell network manager to ignore the eth0 file.

(Edited to include missing "inet".)

Your cable modem is almost certainly restricted to serving IPs via DHCP to a single "learned" MAC address, which is the MAC of the eth1 interface, not the bridge's MAC.

Side note: If you're bridging those ports, then your LAN is directly connected to the WAN, hence not a LAN anymore... Hope you've accounted for this and have appropriate security.