As others say, this would be a really tough time to move a child. When would you realistically be in the new place? Say it was this summer - that would leave whatever was left of year 10 and then two terms of year 11.
Yes, you might find somewhere that did the majority of GCSE's your child is taking - but it is unlikely that the boards and schemes of work would match.
Take English literature - even if a school did the identical books - there are two popular Shakespeare texts, ditto 19th century, tho 20C is more usually the same one. OK, but wil they have done them in the same order? What if your child has done Christmas Carol but the school they move to has done Inspector Calls?
Will history topics be the same? MFL will almost certainly be the same board, but the topics may have been approached in a different order.
Honestly OP is there any way you can delay for 18 months? Your younger child would then move at the end of yr 9, not perfect but OK, especially if the school does a two-year GCSE. Your older child moves in time for sixth form.
If you really have to move, you may find the best option is for your older child to complete a reduced list of GCSEs. What subjects do they study currently? What texts have they done in Eng Lit? MN gets very obsessed with the vital need for 11 grade 8/9s, but in reality 6 GCSEs at grade 5/6 will set them on their way. They would need to do the core subjects and then I guess anything they wanted to do post-16.