B., I see your concerns and totally agree with your decisions. The only thing I would do differently is put her in a after school program. I think it will make a world of difference, as it did for my daughter.
It will cost you of course, but the programs are usually cheaper than a tutor and they are meant for a portion of work daily as opposed to a concept once or twice a week.
I have posted on here several times that my daughter was struggling and is now ahead in reading. We put her in Kumon. I suggest that you research the programs in your area look at the reviews. I have read in some cases, Kumon does not work for their children, but many of the kids I see go in the door without hesitation. Many of them walk our as proud as can be.
They likely will not teach common core, but we all know that there 6 addition factors to get to 10 and it can be argues that there are 11. The way I see it, if she has the basics down, her brain has the ability to focus on the new concepts. My daughter does reading, but one 2nd grade friend in the math program is multiplying and the 3rd grade friend is dividing, both with ease.
Do your research and then schedule a consultation with them.
Best wishes.