It is impossible to assess my level of cooperation based on a singular situation.
Do you think my EX should just agree to my school district?
No I do not. You have a 50/50 timeshare and joint decision making. I think that the decision should be made jointly, based on school quality and best convenience for both parties. I think that the decision should be made fairly. That is how a judge would decide.
I don't know how your timeshare is divided, but if either of you have more school days than the other, and the schools are relatively similar in quality, some judges would tend to go with the district of the parent with more school days. If you have the same number of school days, then a judge is going to tend to look at the quality of the schools, and whether or not one school or the other's location, schedule etc., would cause an undue hardship to the other parent.
I will give you an example of a real life case. 50/50 timeshare. Mom enrolled the child in a very good school that was not the child's regular school, but an alternate one. What would have been the regular school, in that district, was not a very good one. The two parents lived 3 miles apart, but the school was 5 miles in the opposite direction from Dad.
At first Dad did not make a stink about it because it was a good school. However, it soon proved to be a serious problem because there was no easy route to get to that school, so it took much longer than what 8 miles would normally indicate. In addition the starting and ending time for that school, made it totally impossible to get that child, plus the other children in his home to school on time, and picked up on time.
Mom refused to give in, because she took a hard line that the child needed to go to school in her district. So, Dad had to take it to court. What dad was able to demonstrate, was that the regular school in his district was a decent school, and that the bus stop by his house was on mom's way to work, and that the timing of the school schedule made it very easy for mom to drop the child off and pick the child up at the bus stop.
So, the judge had to choose between a school that was a good school, but totally inconvenient for Dad, and a decent school that caused zero hardship for both parents. Dad of course won on that one. Mom eventually grudgingly agreed that it was better. The other school was a bit inconvenient for mom too, but she was being stubborn at the time.
By the way, this story is about my grandson's father and his child with an ex. My daughter, my grandson's father and my grandkids live with me full time and his other kids are here 50% of the time, so I had a front row view to the whole situation.