You might be in a situation where that BT router has it's modem built in - are you using DSL for example? If this is the case, I would have thought (although I have no experience of this, so please correct me if I am wrong somebody) that DHCP would need to be disabled and you'd have to set up a static IP or something for eth0 on the core.
DHCP can be enabled on the external network. The modem/router can provide the extenal nic (eth0) with an IP address and LMCE is setup to request an IP address from the external network. (Can be made static if you wish)
The internal network, (eth1) LMCE makes it's own IP addresses and this is the standard 192.168.80.1 which is best left alone. LMCe then acts as the DHCP server and provides IP addresses to everything else.
Hope that doesn't throw more mud at the issue.
Then, switch things on one by one, in order (starting from the wall), waiting 30 seconds between each
That is how I go from here.
1) The BT connects to the internet and to the Cores external nic. BT is setup to provide DHCP
2) Nothing is connected to the internal NIC for now
3) Install LMCe but stop before running the LMCEinstall script on the kubuntu desktop.
4) make sure internet is working. If not let us know as that is what needs to be sorted.