You can find the admissions policy
at the bottom of this page. I don't believe there is any requirement to move primary school, but of course if your child doesn't it may raise questions about the move. It would depend where you were moving from.
There is information on moving & address evidence
here. This details the evidence you need to give of your move, and the actions taken if BCC consider it to be a fraudulent move.
Please also think about whether there are places available at a local primary school and whether there is another school you would be happy with if your child doesn't qualify for a grammar school. Presumably you don't plan to move between 18th & 31st October? That would be a little tight.
ETA: I've just had a look at the admissions policy for the school & it says that for applicants to be considered as being gin catchment they will must be able to provide evidence of
continuous residency sice October 31st. It could be argued that being at a primary school some distance away goes against this.