Whilst the protesters have been accused of preventing public access to the Cathedral, I recently read a letter to the press, presenting a different viewpoint. According to this letter, St Paul's admission prices are £14.50 per adult, and £34.50 for a family of 2+2. On top of your train fare to London, does this this in itself present a barrier to access?
In contrast, whilst taking a short break in Stamford recently, we took the train to Peterborough and had a very cheap visit to its magnificent cathedral.
Photography permits are priced at £3 but you can choose, as I did, not to take internal photographs.
External photography is free and admission is by donation, so your donation can be in line with your circumstances.
This way, everyone has access.
Sadly, I didn't check out the coffee and cookies available in the tea shop, as we had just had lunch.
Peterborough's city centre is surprisingly attractive, with a large open square (Cathedral Square) that has quite a continental feel to it.
The railway station doesn't have quite the architectural appeal of nearby Stamford, though.