DD dad and 1 aren't together. We have a casual agreement for who DD stays with and when (I.e not through court/ legally). She will stay with him around my work (could be 4 days one week or could be 2 days the next- I can be working from home/in office/away to meetings in other cities so very varied). We get along well and this arrangement has always worked.
It would make sense for her to go to a school in exDP area because due to my shifts I generally wouldn't be able to do the school run. He could take her any day and I probably could 2 days out of 5.
How does it work when applying to a school. He is on the birth certificate but all DD things (passport, doctors surgery, tax credits etc) are linked with my address.
The advice if you were in England would be to ring the council and ask them to clarify which address you should apply from.
The main principle is that you don't get to choose which is easiest for the school run or closest to the best schools, you have to stick to the definition of the child's home.
For most councils this is defined as the address that the child sleeps at most weekdays. Where that is exactly split (it isn't in many cases as most shared care includes one parent having more weekdays) then tiebreakers like where Child Benefit is paid or which address is registered with the child's GP is used to decide.
The reason for checking now and getting clarification is that you don't want to be in the position where you are potentially accused of having used a fraudulent address in the sense that it isn't defined as her home for school purposes. If that happens, it can lead to your application not being valid so it is important to get an answer from the admissions authority before you apply even if it may not be the answer you want.