>>>BT are asking £129.00 to come and repair whatever the fault is
No they're not. To avoid being called out when a fault is within a subscriber's home (rather than within the phone network) OpenReach levy a charge of £129 when they attend such faults. To be fair to BT/OpenReach, they go to considerable trouble (both on their website and when you phone them) to tell people about the charge, so that they then take great care to check that the fault really is on BT's side, rather than with their own equipment.
So, if OpenReach's automated line-checking system doesn't detect a fault, it's almost impossible to convince BT's system that someone should be sent out. (Every time you try to do it online you get a warning that it might cost £129). However if you're fully confident that the phone works on another line (and preferably also that another phone doesn't work on the faulty line), OpenReach will attend and fix the fault free of charge.
All of the above is written from recent personal experience!