I find it strange you didn't work this out before moving in together. You knew the move would lose you benefits but that you were not in a position / preferred not to take on extra work, because you're caring for the (his) baby.
So how did he propose to make the lost benefits up to you? To cover his part of the childcare costs for the care you are providing? To make this arrangement affordable for you?
It sounds as though he expects you to subsidise him and provide free childcare!
The reason you lose benefits is that you are assumed to be benefiting from your partner's income. He is not supposed to be leeching off you by making you pay half of everything, while failing to compensate you for your childcare hours.
I think the obvious answer is a joint family budget for everything. Work out all your income, by writing yours and his into one spreadsheet and totalling it by the year and month.
Then, on a separate spreasheet, all your outgoings, jointly and separately, by the year, month and week (some things are annual expenses, like insurance, others are monthly bills, some might be weekly). Don't forget occasional things; like Christmas, birthdays, other gifts, dentist, sight tests, glasses, haircuts etc. You don't need to get into the minutiae of personal spending together but do need to get an idea of this separately, so you know the totals.
The first thing is to make sure your total monthly income exceeds your essential outgoings. Then see how much is left, for discretionary spending and saving.
You certainly need to allocate equal amounts of personal spending money (and be equitable about what this is supposed to cover e.g. if one of you wears glasses and the other doesn't, I'd argue that expense might be considered part of an 'essential family health' line in the budget, rather than meaning that person has less for personal spending on clothes, nights out and hobbies). If there's anything left for savings, agree whether those are joint e.g. towards a house deposit, or whether they are personal. Set up savings account(s) accordingly.
It is so helpful to see it all spelt out in a budget. It shows you, in a really obvious way, what your fixed costs are that cannot be changed and what areas of spending could be cut, or expanded, depending on income and priorities.