As somebody who provides bids in to Government I hope I can shed a little bit of light on why the Met Office have not been chosen. While I don't know the BBC's bid process I imagine it will be fairly similar.
The BBC will probably have gathered details of as many potential vendors as possible through the publication of a Request for Information (RFI). The majority of these will then have been asked to complete a Pre-Qualifying Quesitonnaire (PQQ) which provides some more details to the BBC as to what the vendors generally offer and what their commercial side is looking like.
So at this point the BBC will have a short(er) list of potential vendors to which they will have issued an Invitation to Tendor (ITT) - this is essentially the vendors full proposal. This often comes in two parts; one technical and one commercial. The ITT is basically an exam with set questions having to be answered in a maximum number of words.
Each of these will be independently marked by both a technical team and commercial team. Given that Government and the BBC work on 'Value for Money' there will be a technical to commercial weighting of say 70:30 meaning 30% of the marks are for cost and other commercial factors. This weighting would differ depending on the contract and flexibility in budgets etc. Given the BBC are a bit strapped for cash this weighting could be more on the commercial side.
Once each ITT has been marked a Preferred Bidder will be selected and they will go in to further negotiation with the BBC. Time will be given for all potential vendors to challenge the results.
So why didn't the Met Office win? - they were simply charging too much for the services they were offering compared to other vendors. So it's not just that the BBC will have chosen the cheapest option as the cheapest would likely score very low on technical. You have to get the technical:commercial balance right in your proposal.
Again, this is the process used by most Government departments and may not be exactly the same as that used by the BBC but I imagine it will be quite similar.