Clearly BA are playing the technical argument as Dave_C says. Your wife may not have had a confirmed reservation, but the fact that this was due to a mistake by the GLA agent means she was de facto IDB'd.

Claim all your costs back (but, as sunrisegirl points out, 45p/ mile is pushing it. What was your actual fuel bill? How much would a taxi have cost?).

I think they should pay out. I can see why they might be squirming, but these regs are there to punish airlines for inconveniencing passengers. It may have been an honest mistake, but it could have been fixed in GIB.

If it were me, I would accept full reimbursement of my actual costs, and I'd be flexible as to the compo (miles/ vouchers/ cold hard cash).