If you want to look at Bromley, you will need to register for grammars this spring/summer. Look at London borough of Bromley, London borough of Bexley and Kent county council websites. Testing will be in the Autumn.
The grammar for Bromley (boys) is St Olaves. It's a super selective. Likewise for Kent, you would only get super selectives within commuting distance. Depending on where you live, and how well your son does, you may get one of the Bexley Grammars. However, if you do like the look of Bromley, there is a very good boys comprehensive school, called Ravenswood. A very good mixed comprehensive is Darrick Wood. You would have to live quite close to these to get into them.
For your DD, the Bromley girls grammar is Newstead Wood. It's a fantastic school (DD1 is there.) but it is a super selective. Bullers Wood is a very good girls comprehensive, but it could be a bit tricky getting a house close enough to both Bullers Wood and Ravenswood.
In Beckenham (part of Bromley) there is Langley Boys and Langley Girls. The boys school is better, but the girls is good to 16 yrs, after which your DD could move, or maybe you will be back home?
Again, there are good primary schools, but you need to live near them to get a place. Are you religious, at all? If you are, eg Roman Catholic, you may have a chance of getting casual places at a faith school. Check the admissions criteria, as you may want to get proof of church membership from your current church. There are also good CofE schools.
Clare House in Beckenham is good, as are Bickley primary and Scotts Park in Bickley, and Edgebury in Chislehurst. However, you need to live practically next door to get into any of these.
So, if you do choose Bromley, research exactly where you want to live, really well, if you are hoping for a state school.