I have an email from her 2 weeks before the deadline saying she’d “changed her mind” about the school and felt I’d “pressured” her into it. That is not true and there is no evidence of this whatsoever.
This could help your case...worth noting however that you can't tell a court how someone feels. Your Ex could have felt pressured by you for any number of reasons, it doesn't mean you had to be standing over them or pestering them to sign.
I just think, on paper - one school is is a 15 min drive for both of us and one is a 5 min walk for one parent and a 20 min drive for the other. Surely that looks like the better option?
It looks like the better option for you, on the face of it. I assume your Ex's position will be that you only want DC in that school because it's closer to you?
Will you be doing an equal amount of pick ups/drop offs? If you're doing the lion's share, the school being closer to you might be considered a more reasonable choice.
I’m guessing also, that at only 4 yo my DD’s wishes won’t be taken into consideration? Eg. She won’t be asked what school she thinks she’s going to or what school she wants to go to? Because she’s definitely under the impression she’s going to the school near me
Very unlikely at that age. At the end of the day, she's only under that impression because that's the impression you've given her. And its equally possible that she could be saying something completely different to her other parent when you're not around. However if you have evidence of having picked a certain school some time ago, it seems reasonable that she would be expecting to go there.