You are informed about the £130 charge which is only payable only if the fault is with your equipment after the master socket. ie a faulty phone or internal wiring all of which is your responsibilty or if there is damage caused by you to the BT equipment.
There is no charge if the fault is before the master socket.
If you prefer you can get your own engineer out to check your home wiring etc after the master socket, which might be cheaper than getting BT but you would obviously still need to pay him.
The £30 cease charge is only payable if you do not move to another provider. It is in BT's Terms and Conditions that you will have no doubt read before signing up to BT.
If you are so sure that the problem is not on your side of the master socket or with any of the wiring in your property or with your equipment get BT to send out an engineer to repair the fault. As you have already been informed, you will not be charged if the fault is with BT equipment up to the master socket.
Have you carried out any checks of your own. See this video.
EDIT: Sky also charge for sending out an engineer if the fault is found to be with your equipment or wiring.
When anyone reports a fault to BT there are checks that we get them to run through. These checks are there to help us locate where the fault is and to help rule out the customer's internal wiring and equipment as being a possible cause.
It is important that these checks are done because if we do send an engineer and the fault is found to be with the internal wiring or equipment connected to master socket, than a charge would be raised to cover the cost of sending an engineer. If the fault is found on the BT Network then no charge would be raised for sending an engineer. We do need to advise all our customers of the potential charge when booking an engineer
If you take a read here - Engineer Home Improvement Service – information and charges for more information.