if you're unsure, there's an ambulance Victoria checklist online with questions you can ask your private health insurer. if any of the answers are no, you don't have it and will need your own cover. it's a bit of a joke how complicated they make it really.
understand totally re wanting another bub. that's not something that can or should be put off until money gets better. you're definitely not crazy for considering it either, as you say, it's just a case of getting on top of things now and finding that balance between you working and childcare.
sounds like the private health cover is a non negotiable at this stage therefore there must be other corners that can be cut to try and get ahead.
have you any family that you could use for free childcare (like 1 or 2 days per week), even just short term to get ahead a bit?
great that your dh has a work car, that saves lots.