An approach function from Swanwick would involve Band 5 controllers doing a job normally done by Band 1 or 2 controllers - far too expensive.
From using Debden 23cm on Farnborough LARS North, I would say, contrary to what NorthSouth said, it's an excellent LARS radar - you can see traffic on primary almost to touchdown at Wyton about 25nm away,(much better than Stansted 10cm due to it being sited about 400ft+ amsl) and RAE Bedford used their SSR feed. I never used it to look towards Oxford so I can't say how well it sees there; I used to use the 'new' Heathrow 10cm in this area as it gave better low cover than the Heathrow 23cm.
Brize already have (or had) a procedure where they would vector for Oxford, but as said before, they tend to close radar positions late afternoon; they may well have sufficient room to allocate a console for Oxford using an Oxford controller. Cranfield Radar used to be done from Bedford, and the initial plan for Farnborough under TAG control was for Farnborough Approach to be done from Odiham by Farnborough controllers, (with the RAF taking over the LARS - we actually started training them for this)so there are precedents.
The main problem with using RAF units for civil tasks is security; during certain practice alerts, RAF stations are made impenetrable to outsiders and to get to the tower at Brize, you not only do you need to access the station but you would also need to go 'airside' to cross the airfield.
All things considered, I reckon the most viable option (not necessarily the cheapest) would be a new primary radar at Oxford, with an SSR feed from Heathrow or Clee Hill. Alternatively, get together with Cranfield and find a site for a primary radar convenient to serve both airfields.