You may be lucky depending on the fare you have purchased - the conditions for my LHR-BNE ticket appear to allow upgrade to a higher cabin (but not to a higher fare in the same cabin) without change fee:
-AT ANYTIME WITHIN THE TICKET VALIDITY FARES CAN
BE UPGRADED TO ANY FARE IN A HIGHER CABIN
RECALCULATED FROM POINT OF ORIGIN PROVIDED THE
CONDITIONS OF THE HIGHER CABIN FARE HAVE BEEN
MET. REISSUE FEE WILL BE WAIVED IN THIS CASE. IF
THE UPGRADED TICKET IS SUBSEQUENTLY CANCELLED THE
ORIGINAL PENALTY WILL APPLY.
Probably worth giving the call centre a try before ruling it out.