Firstly, your childcare arrangements and how much they are or are not costing you are not your employer's concern.
You have the right to request flexible working, they must consider it, and give a valid business reason for refusal. If and when an arrangement is agreed, what you do about your childcare is nothing to do with them, that's up to you. Flexible working requests are agreed (or not) based on whether it works for the business, not whether you can get free childcare.
I appreciate that you are in a tough situation financially, as are a lot of people at the moment. Most people don't get any free childcare, or might get some occasionally if they are lucky, but not all. Most people who need to work are not in the position to announce that they 'don't want anyone other than family' to care for their child. That statement is something most people just don't have the luxury to make.
2 days one week and 3 days the next week seems fairly flexible to me. From your comment that if you had to use a nursery you would have to pay for weekly care but you would only use it fortnightly, I am deducing that your employer is not making you work Friday every week, but every other week. That really doesn't seem unreasonable.
So it's one day a fortnight that's an issue. As your DP is not working at present that should present no immediate problem, so you can go back to work and while he is not working, use that opportunity to explore different childcare options, a childminder for example.