Look at the total numbers doing maths. One big advantage of DS' school was they had so many taking maths that they could set, thus allowing having the advantage (in his view) of not having the super-clever mathematicians in his set. They were also able to reset after AS so those retaking a module or two were in a separate set. Different modules (mechanics, statistics etc) were taught by different teachers.
Consider consistency of teaching. Good maths teachers are hard to recruit. (Some may be good mathematicians or good teachers, it's finding those that can do both which is he issue at A level, which is presumably why a lot of schools don't offer double maths.) There are certainly some weak teachers in the Private sector. However schools recruiting overseas students into the sixth form will live and die by their results so may need to ensure that the teaching at this level is as it should be. Equally if the state school has a big sixth form, strong sciences and lots taking maths this may be the better option. For sciences co-ed can be better than single sex as more boys will probably take STEM subjects.
If your daughter is taking STEM subjects she is likely to encounter plenty of O'Seas/EU students at University. Great to discover at school that whilst O'Seas students may typically work harder (which in itself is a good influence) they are not all intimidatingly bright. And indeed once you get to know people in your class there is a good mix of personalities and some good potential friends.
The main task of private schools, especially selective ones, is to get their students onto the "next stage". They should have lots of experience of sending students to RG Universities. You should expect good advice to both parents and students on subject/University choice and Personal Statements. Really importantly they should be on the ball on results day, helping students who have missed their offers. If offered a place and having a choice to make, ask them about what they do.
If you daughter takes double maths through to A2 and wants to study a maths based subject and has to take STEP, you want to know what help they will give.
Allow for a certain amount of disruption for a change in school. It is quite a lot to be in a new environment as well as dealing with the step up from GCSE. Great for a girl who is ready for a move. But if she is well known and liked at her current school and on track to be Head Girl or hockey captain, she might get more out of staying where she is. Remember either school may start writing its school report for UCAS at the end of the summer term. Not great if you don't hit the ground running and therefore are struggling to be noticed.
Friendship groups reshuffle at sixth form with the scientists spending more time together. If there is a reasonable intake at this point, integration won't be a problem and you don't need to mix with others from your previous school. I assume also that this is partly why the school is offering incentives to non-Asian new entrants, as this should help overall integration.
First step is to look closely at the results and destinations for those taking sciences. Add some weighting for the selectiveness of the intake. If both are getting good students to where they ought to go, try for both and then ask questions.