Although, I will admit that genuine appreciation would be wonderful.
This, in a nutshell, is going to take attitude adjustments on both of your parts. There's truth in what's happened for both of you. She's right and you are right. So, where do you go from here? You say you're not looking to divorce over it. Then, sir, if that's what you want, to make this work, then you're going to have to change. She does too, but I'll get to that.
I think both of you have to quit looking at Child Support as "income." It's not, or taxes would be assessed on it. It is an ex-spouse's contribution to help raise his/her children in having a roof over their heads, clothes on their backs and food in their bellies. The custodian of that money has to use it wisely to ensure the children are taken care of.
Here's where re-marriage complicates the usage of CS. From your posting, she provides for ALL of the groceries to feed seven kids and two adults. I have to imagine that's quite expensive. I'll bet you're always running out of something, milk, bread, apples for lunches, it's always something. I know, been there, done that. But it doesn't seems like you appreciate that. At all. So take another look. She pays for other expenses too. In essence, money is taken from HER x and given to you and your kids. When it's laid out in THOSE terms, it's not exactly fair, either.
She worked 3-4 evenings a week and we decided she would stay at home to be with her kids more.
It sounds like you have buyer's remorse on that one. If money is the issue and it isn't stretching as far as it should, she needs to be working. Of course, that comes with it's own set of problems but it does solve the money problem. At that time you agreed to be the sole supporter of the household. Things have changed and now your health depends on her help.
The thing is, she HAS been helping. It's just not good enough any more. So, you're asking her to dip into the Child Support to ease financial woes. And she's balking at that. It's a stalemate now, finite money coming in with ballooning expenses. How do you fix that? You know. More money needs to be brought it OR lessen some of the expenses.
So, I ask you, in all seriousness, why DO you have an EIGHT BEDROOM house? You had a game plan, somehow it went by the wayside. Maybe it's time to revisit renting out the basement. Kids having their own bedroom is nice but...they CAN survive by sharing some space.
You do realize that almost 80% of second marriages fail. Usually it's one of two things or both: money and kids. That's why your counselor suggested cutting your losses now. Although it's painful, that IS the reality. if you don't want that happening, then both of you need to change the path you're going down.