Groundhog - it's only in the airline's interest to let you switch if one of the following happens
1) Later flight reaches maximum overbooking level at least an hour before the earlier flight is due to depart
2) More people than there are seats are expected to turn up for the later flight
Just because the later flight commands a higher fare, does not mean that it's worth letting you switch. If by letting you switch flights, the later flight ends up with one of more empty seats, the airline has not gained but has lost potential revenue by
a) not forcing you to pay up some cash to switch flights
b) weakened the airline's public claim that higher priced tickets are worth more because they can be changed, meaning you and your family no longer see reason to spend on a changeable ticket when a non-changeable ticket is cheaper.
Ignoring the BA / travel agent issue, if I were working for AC I'd give you a flat "Sorry sir / madam, if you want to change flights you must pay the change fee" unless one of conditions 1) or 2) held