You need to phone BT again and specifically ask that your line is checked at the exchange before you agree to anything else. Also ask that the engineer who checks at the exchange, phones you back with the results the same day. With any luck, they'll pick up the noisy line there and effectively admit the fault is in their equipment.
They can't do these checks from a call centre despite what they tell you. Furthermore, BT don't always do this check before they turn up at your property preferring to bill you if there is the slightest reason to attribute the fault to your equipment. This is why the call centres prefer to give you the standard warnings about the charge.
If they admit the fault is theirs and want to call to check the Master Socket, don't leave the engineer over the doorstep until he says there won't be a charge. Confirm it with him at the front door when he leaves.
What most people don't know about this is that when the engineer sits in his van and submits his report electronically on his tablet, there are two buttons on the screen. One says "chargeable repair" and the other says "non-chargeable repair". It's the engineer's decision alone that will decide if you get a bill for the repair, but you are in a position to dispute an unjustified bill if you've been told that there was no charge.
Personally, I'd go over to another telecoms provider because although companies like Sky and TalkTalk give you the same warning about the charge and you still get a visit from a BT engineer, you can challenge a bill with your provider ( they'll have been charged by BT ) and you stand a better chance of getting the bill cancelled.