People are stating the obviously correct answer, "no," but I think it's more fun to view these as thought experiments to how you would try to get to a yes if your job depended on it. Even though you would, of course, lose in court. And be laughed at (or more likely, lectured at, in the process).

So obviously you have no hope in the lower courts that are constrained to precedent. My assumption here is that we're in a post-Roe world, or else why else would you need to use a novel argument. So we need an argument that will be convincing to at least one of the conservative Justices.

I think my best bet would be the Hobby Lobby decision. In the same way that ruling said the law forced people to "engage in conduct that seriously violates their sincere religious beliefs that life begins at conception," I'd try to also make an argument that there is a religious belief that is equally sincere not only that life does not begin at conception, but that abortion is morally required in certain circumstances.

So you get 4 liberal justices who are pro-choice, and then you try to tell one of the conservative justices that you're a hypocrite if you don't go along in these cases where a person's religion says this medical procedure is necessary in certain circumstances.