The only way one could (validly) argue with that is if it actually wasn’t their position that having the right to birth control is the reason government should provide it.
But I doubt you’ll find many where that’s the case. To ensure against this, simply ask. “You believe government should provide birth control free of charge? Why?”
Once you have your answer, which probably will be something along the lines of “because it’s their right as a woman” or something like that, then you can proceed.
The likely response you’ll get is the default “well that’s different!” When you ask for more explanation you’ll get a bunch of irrelevant nonsense about how birth control doesn’t kill people, or birth control is “necessary” for good health or at least that it’s a wellness product and that you don’t need guns to prevent pregnancy and regulate menstral flow, or some bullshit like that.
It’s very rare that logic will phase someone who holds such beliefs. I think you’re much better off (1) spending time persuading people who are much more inclined to your ideas…basically someone who is just a little more statist than you are, and is interested in the discussion and can reason. Pull them closer to your view of more freedom by opening their eyes to the problems that come with whatever level of statism they support. And in turn, question your own views, and talk to people who support even less statism than you do (if that’s possible).
See here (and definitely follow the link trail from those first two threads in the OP).