I seem to be in a similar position to you in that DP and I live together, and I have my old house I rent out. We have a 2 year old, so fairly recent mat leave.
We do have joint incomes, though he earns more than me. I work less hours too. However, he can only work like he does because I am able to manage the majority of childcare, is this the case for you? If so, it's a reason for him to take on some responsibility for a higher proportion of the bills/ split bills.
With regards to the rental house, that income goes into a seperate account, which covers the mortgage/ house insurance/ repairs/ tax. Anything that's profit (ha!) is joint income.
Are you keeping the profit from the property as sole income, or is it a household income? If it's a household income, then the bills (make sure you include tax in this) are a household bill too, not just private.
With regards to the credit card debt run up on maternity leave- did he offer to take shared parental leave and cover the drop in wages for his portion of that time? As unless he did that, I can't see a reason why you wouldn't consider the costs as joint. Is it more your baby than his? As I've always considered our baby a joint baby, and therefore a shared cost- that definitely includes entertaining and educating them, as the baby classes were doing, or socialising them with meeting other mums.
As I say, we now share finances, were both our wages go into the joint account, and we both have £150 each 'spending money'. If you don't feel this is fair (I do, personally), could you look at doing similar but giving him a little more than you?
One other thing I do is the child benefit goes into my bank account, that is money for extra child stuff. I currently buy her clothes, etc from that, but you could consider using that as a way to pay the credit card.
I know you're getting annoyed at hearing people saying talk to your DP, but you're not going to find a solution unless you do.
Also, you say you're not being financially abused, but if your partner is able to put x away in savings a month, while you would struggle to buy a winter coat without a credit card if you needed one all of a sudden (for example), then I definitely don't consider that a financially healthy relationship. Sorry if that's not nice to hear.