I wonder whether you may be better off asking whether they would be prepared to rebook you directly on to a flight in the very near future that has Club availability, even if that is not today. Otherwise, your best bet might be to take up their offer of flying later today.

A potential problem with them giving you a refund is that - even if you were still at the airport, which possibly you are not - they have no way of knowing whether you engineered a no-show in order to try to avail yourself of flexibility that had already lapsed under the T&C. The airline's best defence against that is to have the passenger standing there in front of them, asking if they (the passenger) can be carried to their destination anyway at the earliest opportunity, regardless of cabin. If someone has missed their flight through no fault of their own, 99% of the time that is what the passenger would want - when something goes wrong, most people's top priority is generally to get to where they have paid to go, not to mess about with champagne and fancy seats. BA has already offered that to you, but you've turned it down. And if BA gives you a refund, it has no practicable way of checking whether you are actually going to rebook yourself, or whether you're just going to take the money (and Avios) and run.

I stress that this is not to suggest that you have engineered the situation on this occasion, but plenty of other people would be perfectly prepared to lie and cheat their way into getting something to which they are not entitled. That's one of the reasons why airlines have policies, and will generally stick to them. Everyone is capable of regarding themselves as a deserving exception, and many live on the basis that rules are only for other people - in which case the rule book might as well be thrown out of the window.