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It's not "childcare" - he is their father, not a childminder.
Whilst your children's thoughts should be taken into consideration, they shouldn't have the burden of making such important - and adult - decisions placed on them. You and their father are the parents and should be the ones making such decisions.
What are the issues your daughter is having with her father? Did they begin before you separated? What is your relationship with him like (As a co-parent)?
If your 10yo said they didn't want to go to school, or wash their hair, or eat their supper, what would you say? This is what the President of the Family Division, Lord Justice Munby, said recently about children who are reluctant to have contact with a non-resident parent, in Re H-B (Contact) [2015] EWCA Civ 389, a case which had run for six years,
[75] …There are many things which they ought to do that children may not want to do or even refuse to do: going to the dentist, going to visit some “boring” elderly relative, going to school, doing homework or sitting an examination, the list is endless. The parent’s job, exercising all their parental skills, techniques and stratagems – which may include use of both the carrot and the stick and, in the case of the older child, reason and argument – is to get the child to do what it does not want to do. That the child’s refusal cannot as such be a justification for parental failure is clear: after all, children whose education or health is prejudiced by parental shortcomings may be taken away from their parents and put into public care.
[76] … what one can reasonably demand – not merely as a matter of law but also and much more fundamentally as a matter of natural parental obligation – is that the parent, by argument, persuasion, cajolement, blandishments, inducements, sanctions (for example, “grounding” or the confiscation of mobile phones, computers or other electronic equipment) or threats falling short of brute force, or by a combination of them, does their level best to ensure compliance. That is what one would expect of a parent whose rebellious teenage child is foolishly refusing to do GCSEs or A-Levels or “dropping out” into a life of drug-fuelled crime. Why should we expect any less of a parent whose rebellious teenage child is refusing to see her father?
Have you considered you, your ex and your children attending family counselling together to try to get to the bottom of the issues troubling your daughter, and work together to resolve those issues, with a view to both children having regular time with Dad.