Going by the numbers available, I suspect that they have direct numbers in the form of 0117 96306xx and 7xx.
Both 0117 963 0600 and 0117 963 0630 have a recording redirecting to the 0871.
BT Directory Enquiries also lists a number for the club shop: 0117 963 0637. I've just called it and it went to voicemail after 20 seconds saying that "mailbox 637" is full and cannot accept any more messages.
If we are lucky, then they may have put the menu on one of the 0117 96306xx/7xx numbers and directed the 0871 to it. I called the 0871 (briefly !!!) and it answers immediately (no rings) with a recording. It takes about 20 seconds before it gets to the menu.
I have found direct numbers for some staff, including Jane Maggs who the website identifies as being the manager of the ticket office. Her number is
0117 9630607This shows that this range of numbers are still used by the club, so maybe the 0871 just routes to one of them.
May I suggest you write/email senior management at the club and express your dislike of the 0871 number. Email addresses are of the form: firstname.lastname@bcfc.co.uk