In that world, the only vote that would be satisfactory would be the amoral vote, because casting a vote consistent with a moral view would be an attempt to 'impose their religion's specific teachings on non-believers.'
That strikes me as something of a false dichotomy.
In this world, someone who claims to be Catholic, but doesn't vote Catholic, ought to be exposed as not being Catholic....i.e., kicked out, so the world knows his claims of Catholicity are faked.
The United Methodist Women gave money (raised through "undesignated giving" )to lobby Bill Clinton to veto the ban on partial birth abortion. Many churches, under pressure from vocal radicals within their membership sponsor "social justice" agendas that have nothing to do with following Christ.
It is one of the strange quirks of Christianity that the Church thrives most when its members are willing to stand up to opposition and share in the suffering of Christ through persecution. In fact, if Christians are not facing persecution in this world, then we are probably not doing our job.
Although I am not a big fan of "bumper sticker theology" it is still a good question: "If you were accused of being a Christian, would there be enough evidence to convict you?"
For men like Thomas More and Deitrich Bonhoeffer, the answer in their times, was "yes."