Unfortunately, sometimes all we can tell you is that you have no "legal" grounds to compel the mother to go along with your wishes. And that's what we are telling you.
If I'm reading your post correctly your child spends a full week with you then a full week with the mother, then back and forth every week.
Well, that's going to have to stop when the child is of school age. Between you and your wife you are going to have to agree on who the child lives with during the week so she can go to school.
You should understand that, unless you can show that your child is suffering egregious harm by living with the mother, the courts WILL default to awarding the mother weekday custody.
You won't like this comment, but I suggest you resign yourself to that fact and work it out amicably between the two of you otherwise you will spend a lot of time and money on litigation and only end up with the same result, at the same time putting your daughter through emotional hell because of her parents' hostility.
What's best for your daughter is that there is no hostile litigation between her parents even if you have to make concessions to avoid it.