I agree with Swimming about bags. DD had a "five bag day" which was awful when jostling through a main line station in rush hour.
At minimum you would need to discourage your son from playing a large instrument, taking Art GCSE, playing more than one sport and even then choosing his sport carefully. Rule out any suggestion of playing GK in hockey for example. He would also need to think about getting too involved in drama. Its not just the performances, but lots of rehearsals after school. Driving from Dulwich to Barnes to pick him up every night for a week would be awful.
A school coach, or a tube full of friends, is different, and provides some social and wind down time. On your own can be miserable especially if you are surrounded by a full carriage of commuters. No seat = no chance to do a bit of reading.
Trains and tubes have their issues. The worst problems come with buses, especially if the route is affected by significant road works lasting months.
You might ask SPS about what transport arrangements there are, and how many students they have from your post code. I remember one family years ago, who were looking at a shared taxi arrangement once they realised how miserable it was to rely on two forms of public transport plus the trudge across the river. They decided to move instead!
It also depends on where your younger son is likely to go.
FWIW I would go for Dulwich and if he is really at the top of the year and wanting to study in a more selective environment, think about a move to either Westminster or SPS for sixth form. Dulwich offers an awful lot often, especially sport, to a higher standard than SPS.