While I'm sure that there are Republican politicians who honestly believe that they are doing the Lord's work by pushing their right-wing Christian agenda, I am also certain that there are a lot more who use evangelical rhetoric just to get elected. I'd love to call their bluff. They did lose some credibility all on their own with the recent tragedy in Florida, and the timing might be right for the idea that I am about to present.
I would love to see a senator propose an amendment to the constitution that did two things: define the fetus as a human being and require that the federal government provide a socialized universal health care system. More below...
Most evangelicals don't really care about "socialism." They are focused on moral issues and not on the material issues. Thus, they will not understand why their Republican politicians are in such opposition to an amendment that would clearly lead to the outlaw of abortion. The Republicans and their real agenda would be exposed in a raw way.
At the same time, Democrats could all make their own judgements. Many Democrats are opposed to a single-payer health care system, and others will think that the abortion issue is more important than the health care system issue. It's up to them individually, and Democratic voters can probably forgive each Democrat for their vote in either direction.
As has been discussed before, socialized health care can be a wedge for issue for business anyway. While business is typically conservative, socialized health care would make most American businesses more competitive in the world market. I am curious about what business would do on this issue anyway.
If the amendment were to somehow pass (doubtful,) hey! We'd get universal health care! A lot of women would be hurt by the amendment, but I'm sure that a lot more women would be helped. Of course, everyone will have their own value judgement to make on such a proposal anyway.
If the amendment were to fail (the likely scenario,) policy would not be affected, and the Republican exploitation of evangelicals would be exposed to the religious right.
(It would not be fair if I were not to meantion that I am nominally pro-life. I think that abortion should be illegal, because I am uncertain about the humanity of the fetus and thus its rights. I would rather deny a woman a right to abort than what could be a person's right to live. That said, I cannot support the illegalization of abortion until the following things have been done: the provision of free contraception to all women in America; drastic reform of the adoption system, such that no woman has to keep the burden of an unwanted child; and a thorough sexual education of American children. Needless to say, I'm pretty much pro-choice for the time being and the forseeable future.)