I think you are warring with two separate options here.
#1. You need daycare/after school care for your children, & need to make a decision that is affordable for your family.
#2. You have an opportunity to engage a family member who is currently not working to provide the childcare for you, at an inconvenience to you.
If you look at this strictly by the numbers, the B/G club, daycare & babysitter makes the most sense financially. $425/mo vs. what you are currently paying, which is $20/day for 20 weekdays per month (some months have 1-3 more), plus the time & gas expense to bring your sister to & from your home.
From your SWH, I am suspecting that one of the issues clouding your opinion of having your sister watch the children is that you feel you are doing her a favor by paying her, & she should be grateful for the opportunity. Additionally, the time, inconvenience & gas expenditure appears to be something you are taking into consideration when factoring the cost to you & your husband.
However, what you & your husband need to do is decide which is more important to you: Having family watch your children, regardless of the inconvenience & cost to you both, or having reliable childcare for your family that is affordable and does not incur additional cost or time inconveniences.
In reality, childcare is not cheap. And for 3 children, often times the SAHP puts most or all of their paycheck towards the childcare, in order to hold a job that might bring health insurance, etc.
I think most babysitters make more per hour, due to the demand for babysitting coming during premium hours - evenings & weekends. But even daycare is not terribly affordable, when you are looking for one person to watch only your children. This is why daycare centers can charge less - there are higher ratios of children to staff.
As to whether or not your sister should accept what you & your husband can afford to pay, or not watch your children at all - well, that is certainly her choice to make, not yours. Quite possibly, she would be able to make as much money working part-time hours somewhere close to home. You are paying approximately $400/month - she could make the same amount of money working 10 hours a week for $10/hour.
For as outrageous as it seems that your sister is not grateful for what you are paying her without asking for an increase, it may be equally as outrageous to her that you aren't willing to pay her minimum wage on the hours she does work for you. I notice in your post, you list how much your husband makes per hour. It seems like you are making a comparison - he only makes $XX, therefore, it should be acceptable to only pay $X. However, most (if not all) services do not charge based on how much income the person in need makes.
My best advice to you is: Make a decision you can live with, that works best for your family, & leave the judgement on the table. She is your sister - and if the best way to maintain your relationship is to look for childcare elsewhere, then do so. If the best way to find affordable childcare is to look elsewhere, then do so.
Here's hoping you can make peace with the decision before you, & maintain a healthy relationship with your sister. T.