Could you place a booking on hold for whatever is in policy, then just book whatever class you want to travel in and then claim back for the "held" fare. You'll lose the £10 or whatever BA charge but it'll be cheapest, unless your company would just accept a screenshot of the payment page which I doubt many would.

This does assume that you can book the travel yourself, of course.

Surely cost centres and profit centres aren't a foreign concept?

I work in a profit centre role and my business travel is purely domestic within the US. My company pay for F because the work I can do on the plane generates far more income than it costs to put my in the front seats, work I couldn't realistically do in Y.