Why waste useful BML capacity on that when you will be able to change at Farringdon? Please explain which Brighton to London services you're going to pull to make room.
Then you could also ask the same question with connecting any Heathrow trains to SWT lines, as to which trains would you be cutting?
The purpose of my idea and the train need not start from Brighton, it could in fact start from Gatwick, but I would have it replace one of the GWR trains as far as Redhill, but have it go on the BML and replace an East Croydon - Watford Junction train service to the point of Kensington Olympia.
The GWR trains I would cutback to Redhill, with none going to Gatwick so that people change either at Redhill for Gatwick or when the connection on the Reading side into Heathrow has been completed, then they can change at Heathrow for a train to Gatwick.
The Western route study already includes a likely Western Rail link service to Basingstoke (or further) via Reading, as well as the more obvious GW destinations. I don't think the Southwestern possibility is any less likely than the others, in the way it is written. Direct Southern access to somewhere like Woking or Guildford via the Staines area doesn't cause the same issues the original Airtrack scheme had, which were mostly to do with the routes towards Waterloo. A west facing bay at Staines would tick a lot of boxes without running towards Waterloo.
But you would need to be cutting existing SWT trains to have the capacity to be able to do this change, so as per my question above which trains would you cut to enable this?