Can you not connect the modem that will work on this 'iffy' line to the other router/modem combo ? ( or not all the right connectors ?)- it seems to me 'mad' to have a router & then not use it as a 'buffer' to the outside world & start using ics ..

btw if the ISP is providing the BT modem, perhaps given the nature of this, they might be persuaded to swap that out for something that can connect to the router, admittedly this seems to ignore the real problem, but I appreciate the 'further cost' implication