It's a difficult decision to make: we had one son who did transfer to CHB from Arden for sixth form (it was at the stage when the Arden 6th form was only just starting so his other option would have been Solihull school , with a scholarship, as he was not keen on SSFC). He thrived there, though he tells me it was touch & go for a week or two adjusting to the very different atmosphere. He particularly enjoyed the music, as he was doing music A level in a small mixed group, and also joined many ensembles and the BSSO and made several good new friends. it certainly helped him hugely in his Oxford application
Others we know, equally bright, have lasted only a few weeks at grammar 6th forms before transferring back to their previous school (which has usually been happy to reintegrate them). Others have transferred, succeeded academically but never quite felt part of the new school (I've heard this particularly with girls going to CHG and to Shottery).
Our daughter was adamant, despite having an offered place at CHG, that she wanted to stay at Arden: despite OH's dire warnings she got her AAA and place at Manchester. Her argument was mainly to do with the travel time and with her much preferring a mixed-gender environment and not having to start from scratch with a completely new set of friends & teachers etc: - also a very relevant point when it comes to writing UCAS references etc where a new school won't have a head of year who perhaps has known a pupil for several years (though this could also be an advantage of a move if a clean slate is needed for whatever reason). If KEFW had not been so ridiculously far away it would have been an attractive option, but Kings Heath is quite far enough from us when Arden is within walking distance!
Sometimes a move is the only way to achieve a particular combination of A level options: for some the detailed choices are far less important than the environment where they feel secure and able to work. Each child's circumstances are different