BACF has just trimmed its out stations and ditched most or many of its newish summer sun flights out of BRS STN and BHX
They have ditched BRS and BHX, but I think this was more to do with ensuring the operational resilience of the core LCY ops, and reducing operational complexity and resultant costs.
I think a BA-branded op would go down very well at SOU, provided prices keep the user-base on-side. Doubtless it would be a much trimmed-down/core routes only offer, but the ability to sell flights through BA.com could lead to an upside in terms of viable ops to more European capitals/centres of commerce? And provide a shot-in-the-arm to the sun routes also?