Short term move should not prove too difficult.
BA can reduce down from 4 desks - they don't need that many with the % of pax using self checkin or on line checkin. Use some 'hand baggage only' desks in the new hall - even if it means splitting check-in for individual carriers and flights. Some common bag drop desks and a bit of sense in opening/close times and you've solved the issue.
If you look back and see the volumes that used to go through the 'old' hall before the extention then it is do-able.
The question is - who within the airport company has the experience and nouse to make it happen?