"Removing the faceplate" could be one of two things, 1 is removing a standard faceplate in which case the extension was probably fitted incorrectly if not fitted by BT. 2 is removing the bottom of the plate only, if your telephone plate had a split in it running left to right above the screws and when the bottom was removed the extension was dead then the extension was not the issue. The plate in this case would look like this:
Master socket with test section
If you had an internal extension not wired by BT and that was causing an issue then they are within their rights to charge the fee (as they should have mentioned during the troubleshooting phase of the call), if however the issue was not internal then they cannot charge you. So the questions here are:
do you have a split socket on your line,
were you asked to remove the bottom half of this and plug into the test socket,
if you were asked and did this did the extension lose it's dial tone when you went in to the test socket?