The problem for those who want to call themselves "conservative Christians" is that they claim a factional Christianity. And that's a major problem from the outset. Any modifier attached to "Christianity" is heresy in the classical sense: a "dividing," as if there were a "conservative" Christ somehow different from Christ.
There's the problem too when any such modifier is a human categorization. "Conservative," from the realm of human politics, is certainly that. Regarding Christ as if He were subject to human categories is ultimately denial of His divinity, and His sovereignty. If Christ is subject to our categories, who then is the master ? Those who would reduce Christ to their human categories do so to make Him serve THEIR purposes.
But "Conservative Christians" typically claim that they must take that label to differentiate themselves from "liberal" Christians. They protest that they don't want to be associated with those who deny Christ's divinity, virgin birth, atoning death, etc. They claim they want to make it clear that THEY, in contrast, are "real" Christians (with all the pridefulness that implies).
I'm always surprised by that argument. Modifiers are only necessary to distinguish between things of the same kind: "green apples" and "red apples," for example. My "conservative" friends' insistence on that modifier indicates they must consider (those whom they call) "liberals" are another kind of Christian.
Christianity 101 for "conservative Christians:" no one who denies Christ's divinity and the biblical facts of His life is a Christian. The honest distinction is not between Christians who are "liberal" and Christians who are "conservative:" it's between Christians and non-Christians. There are ONLY "real Christians" and those who really aren't.
Another problem of "conservative Christianity" is its belief that Christianity is not quite sufficient. Why else would we add something to Christianity, unless we believed Christ didn't cover everything needful ? More to the point, why would we consider that He missed giving us a necessary political thought-system ?
Joining "conservative" to "Christian" clearly indicates a belief that Christianity doesn't adequately address human political ideas or behavior. Such a belief...or rather, lack of CHRISTIAN belief...evidences "conservatives" deep ignorance of Christian teaching. Christianity 101 for "conservative Christians:" study Romans 13: 1-7, and prayerfully follow those teachings.
Believing Christians, in contrast, are rooted in the fact Christ's teachings ARE all-sufficient, and address all our questions and needs. That's the substance of Christians' confession that Jesus is Lord, and sovereign over mankind and all man's works. Christians believe that Jesus is Lord even over human political acts and ideas, and has given us His full counsel towards both.
"Conservative Christians" purport that Christianity must be supplemented by a human poltical belief-system. But if Christianity is true, that Jesus and His teaching ARE sufficient, "conservative Christians" deceive themselves. What is perfect (in biblical terminology, "complete") is a unity in and of itself. Discarding, or re-fitting, or replacing components of what's whole and complete doesn't make it work better.
A unity can only be accepted as a whole, or rejected as a whole. It's false that Christ left us no instruction towards politics: and disingenuous (if not outright dishonest) that any human belief-system can be made to meld with, or supplement Christianity. God's thoughts are not man's thoughts. The two are mutually-exclusive: and the human belief-system of "conservative Christianity" can only be adopted by dispensing with "the mind that is in Christ."
God is merciful. I can pray (and do) that he'll extend His grace to "conservatives" for honest repentance. If "conservatives" will yet see that their prideful factionalism rejects Christ in favor of their "own way," perhaps some will turn back, and follow and obey Christ.