Why should the OP's ex's preferences for how their child is educated trump the OP's?
Why should the OPs preference trump the exs?
Based upon personal experience of discussions / disagreements about schooling, consensus and respect for each other's opinion will stand your DD in good stead in the long-term.
Whether home schooling or conventional, the most important thing is your DD's happiness, so if you can find a way to get through the impasse without going to court, you'll be better (and less light of pocket).
Also are your intentions for home-schooling just when your DD is in primary education, or into secondary? Maybe explore those concerns together, to clarify what you do and don't want.
Maybe both of you write out a brief list of pro's and con's for both options (including specific alternative schools) and if you can talk it through, good. If not, could you try a mediation service outside court, who can help.
I think it may be a burden to your DD to be put in a position of expressing a preference - what if she expresses her (childlike) view of the world which doesn't end up being in her best interests. It just adds a further layer of complexity doesn't it? Fine if she was in secondary school, but she is still very young...
I think the only way to make it fair is that the parent who has the most time with the child gets to decide
It isn't about the parents, nor about "fair", surely? Isn't it DD's needs they should focus on?