I don't think republican care about abortion itself, they just realize that they're on the wrong side of most issues they do care about and it just so happen that the church is also on the wrong side of most issues, so they're natural allies. Plus very religious people often care about religion more than absolutely anything, so they're easy to manipulate by draping your message in religious connotation, while not having to do anything religious yourself (see Trump being popular with the most religious people despite being pretty much the exact opposite of everything religious pretend to be). It help that they can plaster picture of baby on everything.
Even the church probably don't care that much about abortion itself, its more about rallying people to their cause using an emotional issues (and hoping they can convert and have them give donation in the process).
At this point both the republican and the church have made so much noise about abortion that walking back, or even failing to deliver changes, would be seen as failure. So they'll keep pushing the issue no matter what. In that sense there's no negotiation possible. And they'll ignore every arguments and statistics about it, if they accepted them it would be admitting that they were wrong on the subjects and wasted their time and energy.
The only way forward would be to find a way to de politicize the issue, but the democrat also use the issue for electoral gain so I doubt that will happen. Ideally you need to change the message from "How dare you oppose this thing that I approve of!" to "Wait you oppose this?! How weird, I bet you think the earth is flat too.".