I certainly don't think it's morally wrong and I think you should do whatever works best for your family, especially if it will alleviate mental health concerns. But considering you and your partner only work 7 days between you, you have only one 2-year-old, and you have local family support, it sounds like you may have a time management problem more than anything. (I've been there.) If that's the case, more childcare won't relieve your sense of overwhelm and will in fact just make you feel guilty. (Like I said, been there.).
Can you figure out exactly what tasks you are having a hard time keeping up with, and where your time is going? Many things can be done with one 2-year-old in tow. Your child doesn't need your constant attention and there's no shame in using CBeebies for an hour so you can zone out on mumsnet make dinner or pay bills. Take a hard look at where your free time is going. There are even apps to help you track this sort of thing. Between you and your partner you should be able to work out a better plan to stay on top of things. If it is a matter of housekeeping, can you do a huge weekend purge/organising session to simplify things? Can you batch cook meals on the weekends? There are loads of strategies that can help but first you have to identify the areas that are causing the most stress.
I just know that when I got more childcare and didn't get more done, I felt even worse. At least if I'm not getting stuff done and I'm with my kids, I can feel good about having the time with them. In retrospect I should have tried harder to be more efficient before just assuming I needed more time. Also you just have to accept that there is a certain amount of "insanity" that goes along with family life, especially with two working parents. I don't think any mum I know ever feels completely on top of things. Are you setting impossibly high standards for yourself?
All of that said, I may very well be inappropriately projecting my own experiences onto you. But I'm not saying this is rational, and you should absolutely NOT feel guilty, about the tax credits or anything else, if you need the extra time and it will genuinely help you to feel better and healthier. We're all doing the best we can.