Write a snail mail personally addressed to either the CEO or the Chairman. That will get your case passed to a better class of problem solver who will stay with your case until it is resolved.
Or, find someone who is employed by BT. (Not a contractor). BT staff and pensionners have access to a better class of problem solver.
The more you can convince them to completely replace stuff like their boxes and cables, the better. (Assuming the engineer knows what s/he is doing)
The "Engineers" they send out are, er, variable.