I had a similar situation - we have linked expenses to a corporate card so you can't separate your own spend with actual transactions easily. In fact our bookings are also usually done centrally. So what I did was suggest I'd beat the price quoted by the travel department and used enough avios to get it down below the economy fare plus booking fee, didn't take much. Then I claimed for that amount with a receipt which was a screenshot of the BA payment receipt which doesn't mention travel class.
You shouldn't have to play these games, I personally would gladly pay the difference between what I can get from the travel people and what I can book myself. But that's corporate culture these days, there's a perceived virtue in suffering when travelling.