For a 3yo, staying at home learning is the best! If your child is an extrovert who enjoys the company of other kids, perhaps daycare for a few hours daily for 3x a week will work for you? U can get some time-off yourself too. Another option is for u to homeschool her and give her some enrichment programs outside.

Montessori didn't work for my ds too. He was frustrated that he had to do the presentation from step 1 to step 20, and to repeat them till perfection. He wasn't interested in being with other kids too (although they were of mixed ages). Another reason was he prefers staying at home and do things that he likes .. read, play with lego, being with me etc. Even now at 8, if he has a choice, he would stay home!

Sorry not much help here. I hope u find something that suits u and your dd soon.

I am enrolling my dd2.3 in a preschool which only accepts kids who are 2.5/80% potty trained, so she'll start in a couple of months. They are not focused on academics, but will do the singing & reading, and they have a huge shady playground. They do full and 1/2 days. It is a church school, though we don't attend that particular church. You might check out some of these as they are usually low cost and typically don't require membership in the parish. (I did inquire about the level/content of religious instruction to make sure it would not conflict with what I would want for her.)
As for flexibility they didn't seem opposed to moving a child around to whatever class seemed best...

My oldest DD started pre-school a year early. She turned 3 in Dec., so I started her after Christmas break.

Technically, she was three so the school allowed it.
I called the director prior and said things like:
There is a new baby at home and it would be so good for her if she had something that was just for her.
She is very mature and gets along easily with other children.
Could we try it for a month, if it doesn't work, just call me, we'll take her back out no problem.

Of course it worked. But the director had an out if it was a disaster, because for some kids it might have been.