The Flyer is operated for the airport by First Coaches. It's been running for over ten years with dedicated, airport-liveried 50-seat coaches.
The Flyer is more than a bus service. Because BRS has no rail connection, and is unlikely ever to get one, the Flyer is part of an integrated public transport system to the airport and it's possible to book through rail tickets to the airport from most stations in the country, through National Express tickets and through local bus tickets, with the last leg (first leg for inbound flights) on each using the Flyer.
I would be extremely surprised if the airport hasn't ensured the new buses will be for its exclusive use in its contract with First.
It cannot afford to have First playing fast and loose with the type of vehicle it uses.