A childminder can choose who they look after.....but having agreed to take a child, they owe them and the parents a decent time period of looking after them, before saying they don't want to/can't look after that child. It is unprofessional to take a child and bin them after 3 days.
One of the CMs responsibilities is to know enough about a child before they start caring for them, so they know if they can cope. If a child has any extra needs a parent needs to make that clear. Every CM would expect a new child to be a bit unsettled and is needed for them to settle.
It looks to me as if you were binned because she found someone else she would rather have....perhaps more hours or whatever. Having agreed to look after a child, it isn't illegal to do what she did, but it is immoral to make an agreement as a childminder without having any sense of commitment to that family - being so prepared to renege on the agreement as soon as another option appears. If someone else appeared who she would prefer to look after, give more hours etc, she could say to you that she would look after your child for 3 months, but would then need to stop. This would be disappointing, but reasonable.
The money aside, this is awful and has left you in a hole regarding childcare now.....but long term, you are well rid of her....she isn't the kind of CM you want. In the long term you'll look back and be glad your child didn't spend months with her.
CM can be a brilliant service for parents and a brilliant work option for many. The nature of the individual agreements does mean that many people don't have in place proper paperwork laying out what is expected and required for both sides regarding agreements made months before the start date, deposits, retainers, fees, holidays, notice periods etc. Both parents and CM are known to swing the lead and break agreements (even if written down) and to let each other down. When it happens, there's not an awful lot the other party can do....even if legal action can be taken, it rarely is, due to the bother and potential expense.
CMs can be brilliant. They can offer something that nurseries simply can't in terms of home care and a personal service totally tailored to the times you need. It's so important to get as strong a sense of the potential CM as possible to gauge if they are likely to let you down. You can never know for sure, but taking up references, visiting several times and looking at their paperwork and the contracts they offer helps you judge. Anyone who isn't happy to see you a few times, who doesn't ask in depth questions about your child, who can't give you previous customers to contact for references, and who doesn't offer clear paperwork which covers all the areas I mention above should be steered clear of....because there is just too much uncertainty.
OP, you've had a bad experience, a CM could still be right for you and be the best option. Or a Nursery might suit you more.
I would be clear with that CM that you think their handling of this has been very unprofessional. I would write to her saying that any child takes time to settle and you would expect a registered CM to allow longer than 3 days to make a decision and that in light of that failure, you feel she really ought to return your money.