Brize always use the Cotswold RPS rather than their own QNH; at least they do when they hand traffic to adjacent LARS units.
Mode C is transmitted from the aircraft on 1013 but it's displayed on ATC radars based on whatever pressure setting you input, the conversion being done by each individual units processor. I know Lyneham used to input 1013, but other RAF units usually input QFE, whilst civil units input their own QNH.
There is no such thing as the 'London RPS'; Benson should issue their own QNH as the QNH of any airfield under a TMA is deemed to be the same as any other airfield under that TMA.