We may be slightly ahead of ourselves here. The BBC has said it has a responsibility to put the deal out to tender. This means the current contract with the Met Office will end.
There's nothing, anywhere, to suggest that the Met Office cannot put in a bid for the new contract and nothing to say that when the current contract has come to an end there will be a new contract between the BBC and the Met Office, if the BBC concludes that the Met Office bid offers the best deal. This may, or may not, be determined entirely by price.