I go back to when the London Underground Railway Society had absorbed what was London Roving..and so Guinness Superlatives asked us to supervise what had been up to then two or three records: A, B and C, depending on whether just the tube was used, or public transport between lines, or taxi/private car. I think an all-time record was set by a young chap whose mother ferried him around at termini! Anyhow, back around 1980 LURS was asked to supervise a new attempt which was the first for years. It had been agreed that a new record would be set as the system had changed so much. I did my bit on the District when they went to Turnham Green thence to Richmond and back and then on to Heathrow. They caught the last District at Monument as Desmond Croome asked the District Controller to hold the train there as the team were coming over from Bank. They finished at 01.15 or so and had their photos taken with a BR ticket collector! In those days we had Ongar, and Watford on the Bakerloo, exhibition-only on the Olympia branch and peak hours to Shoreditch. So you started at Ongar at 06.30 I think it was, on a day when there was an exhibition on at Olympia, during the week!
Back on to the query, I am sure that people have boasted that they have 'done' all BR stations in the past. It would surely take weeks? Sounds like a call to GBR is apposite to ask if a new record could be set? If the same criteria as the tube record is applied you need to have stopped at a platform and not passed through it. That means fun with one-a-week services!