I would personally think with the sort of wording used, and the outrageous claim for reimbursement of the flight (which was completed, by road, so you can’t realistically double dip for the flight plus the cost of doing that), anyone looking at it might well be tempted to put it to the bottom of the pile for action.

I can also see that BA might find a reason to refuse this because it’s not an approved method of travel - they are rarely logical about these things given they’d prefer to put people in a hotel for several days rather than pay for re-routing on a different carrier. If I was told to either stay overnight and fly the next morning, or get on a train, that’s exactly what I’d have been doing. Incidentally, a walkup standard class off-peak fare from Newcastle Airport to London is £134 per person - so totalling about the cost of the non-approved car hire. The OP may well have felt he was being generous towards BA and helping out by reducing costs, but it could backfire.