I'd try pulling the power in the Livebox, the relay will switch over to the BT circuit. Call your BT number and check it rings OK. Almost sounds live the DECT unit is doing the answer phone bit. You n ...
Are you into the Livebox config on 192.168.1.1 ?
SIPGATE works ok as does MSN voice calls
Dont think there's anything in the Livebox that would block it