Here's a few points that come to mind.
Corridors in schools are often what used to be called 'dual purpose' areas, i.e. they are circulation spaces and also areas to display students' works of art and to install mini library extensions (no coat racks though, no no no). Are these corridors so used? May they be in the future? It would make a big difference.
English Heritage or other bodies involved in listing buildings usually have no problem, even for the Grade 1 buildings - and I doubt this building is Grade 1, in accepting fire resisting screens and doors that cause virtually no damage to the existing structure (such as a few screw holes) and which are completely removable. Any such doors should, of course, be held open on magnets as previously suggested otherwise they are likely to be destroyed in a short space of time.
Subdividing corridor doors serve a couple of functions that might be important if there are (often, to be expected) fire safety failures in the future. They reduce the chances of both staircases being lost to a single fire - the required fire safety failure here would be doors to staircases wedged open or damaged (not unknown). They also reduce the distance that anyone might have to travel through an unprotected corridor that is affected by smoke from a fire in an adjoining room; and there is no requirement for a fire safety failure here.
The enclosure of each classroom probably will compensate for both of these functions, provided that the corridor is free of any fire load, but it won't future-proof the building against the introduction of fire loading into the corridors. And it will also be a lot more expensive!
Dividing a school into many small compartments might have benefits for protecting property, which is very important in schools, but it sounds like an unusual solution.
This sounds like my old school, it's not in Cotham, is it?
Stu