Agree with PPs about Bromsgrove - they have both A level and IB so options would be open. Maths department is huge and successful, the brightest are definitely pushed in maths. Science is very strong and well resourced but ICT not really a thing there. Opportunities for clubs etc, I'm sure, but not a GCSE.
Most boarders are international (some Brits based abroad but also many foreign students from all over, especially Russia, Germany, and many East European and Asian countries) so it remains busy at weekends, despite having many day pupils. Having local kids also means some boarders get invited out at weekends which is nice.
On the practical side, school-run transport to and from airports at the start and end of half-term and term makes that side of planning easy (and no danger of a younger or less responsible boarder disappearing or missing their bus/taxi!) There are NO exeats at all, which means the overseas kids never get carted off to distant relatives or host families. I think this is quite rare, most schools have exeats.
Global Perspectives is an IGCSE subject, quite popular in international schools but hasn't really taken off in the UK yet from what I've seen. But Bromsgrove (and I'm sure most of the other schools you mention) has Politics at A level / IB.
Sport at Bromsgrove is amazing, it usually ranks top few in the country and although there is a focus on typical British sports like rugby and cricket, the athletics, swimming, basketball etc is all excellent too.
Good luck in your search!