Actually, the guidance is that local authorities shoukd process applications from abroad, but obviously using the home address there would put an applicant way down tge list on distance where that is a deciding factor.
https://www.gov.uk/guidance/schools-admissions-applications-from-overseas-children
Apply from whatever address you are at on October 31st, through the Common Application Form of the local authority where you intend to live. Name your preferred schools in order, with the (preferably catchment - look for the maps on the KE Foundation website) grammar school first, if that would really be your first preference if your DD were to be offered it somewhere down the line), then other schools in the area you are moving to in your true order of preference.
On March 3rd, you will be offered the highest of your preferences that can be met - probably none, but you may be offered one of them first off, if the school isn't oversubscribed. By then, you should have a local address to confirm with the LA, if your move goes ahead in December, which will improve your ranking on the post allocation day waiting lists of schools ranking by distance. With a confirmed local address, the LA will also allocate a place (somewhere) to your DD.
In the meantime, you can sort out how to get your DD to sit the West Midlands exam. The admissions policy for the Handsworth girls' grammar, for example, states
Test registration is made electronically via the relevant website and must be submitted by 4pm on 28 June 2024. Candidates who are not registered by the deadline may only be tested in exceptional circumstances and considered after the main round of allocations has been made in March 2025,
which indicates that it is at least in theory possible. Not being aware of when you were moving to the area until it was too late to register on time would be counted as a reason in our area (but I'm not in Birmingham).
Obviously, you can name more than one grammar school on your CAF if you want, but your DD might be most likely to get a waiting list place at your catchment one, if she manages to get to the point of being eligible to have her application considered.