Hi, my DD is in St James year 8 (started in year 7).
I don't have any knowledge of St Catherine’s though and cannot help with comparing the two.
Regarding St James, we are happy with the school. Quality of teaching is good overall and in certain areas - classical and modern languages, history, science, music, exceptional in my opinion.
It's a small school, only a two form year group, and that has its own advantages and disadvantages. Teachers know all the girls and we found that helps in pastoral support. Friendship issues or any other problems are spotted and handled quickly. It is hard for someone naturally shy or less confident to step back and hide in the crowd; everyone is encouraged to participate in everything. But small size also means certain things are harder - for example forming orchestra with all the girls at right level of skills for the right instruments etc take longer. Also, for such a small school, it seems to do surprisingly well in some sports, like lacrosse, netball and football.
London location works for us and the girls qucikly become comfortable navigating central London, meeting up etc. I like that DD can manage her route in tube etc, and now a days meet up with her friends without any help from us.
Regarding the boys' school, there isn't a lot of interaction. There are few joint events, when the girls go to Ashford for a joint summer concert, or the boys come to London for a church service. But travel time between the two schools make joint events hard. There is bit more interaction in sixth form I believe.
About meditation and vegetarian lunch - I was (am still) a fan and one of the reasons we chose this school over few more high profile ones are its ethos (we also wanted a small school). I believe the quiet time (~ 5 mins pause) between lessons is useful and the vegetarian menu is a good thing. But, DD and her friends are emphatic that quiet time has no benefit and is in fact annoying. Some girls also did not like the veg menu starting year 7, but I hear fewer complaints now a days.
Happy to answer more specific questions if you have any.
There is also a Good School Guide review here if you haven't already read it https://www.goodschoolsguide.co.uk/schools/100107/st-james-senior-girls-school/E690976#sch_tabs-2