Basing this discussion on privacy laws was a backwards chickenshit workaround, in the first place. SCOTUS kicked the can down the road in writing their decision in Roe v. Wade, setting the stage for precisely this. It was never a "good" judgement.
Abortion rights shouldn't be covered under privacy rights at all. "Privacy" shouldn't even come up.
What needs to be passed is a right to "bodily autonomy", or "security of the person"; different terms for the same thing (Canada's Charter of Rights and Freedoms includes the latter). If you want to get some dipshit Republicans to cross the aisle, point out that said right would prevent forced vaccinations or government tracking implants. Sure, the latter's nonsense and the former is irrelevant since it has nothing to do with vaccine mandates, but they don't pay enough attention to notice that; they'd have to explain to their base why they won't protect them from the government forced vaccinations.
Once you've got a right to bodily autonomy recognized, abortion rights are naturally and irrevocably included. In a much stronger sense than Roe v. Wade ever did.
Just frame it as the right to control your own body. Make it wildly generic on that basis. There's little to argue about once you make it that broad.