I think Chris makes great points, and his approach is absolutely solid.
In general, we can divide the steps from Nature to the Church in Three.
Step 1: From Nature to God. (in the first step, we refute atheists, agnostics etc by Philosophy, Natural Law, showing God is Supreme Goodness etc)
When it is proved God is Supreme and One, ipso facto it follows that all polytheistic religions, howsoever many they may be, are false.
Step 2: From God to Christ. (in the second step, we refute Jews, Muslims etc, who confess one God, but not Christ, from Prophesies, like the Messianic Prophesies admitted by Jews; from Public Miracles like those associated with the Resurrection of Christ and the effects it has left in the lives of His Apostles in history, the Shroud of Turin; those from Our Lady in Gaudalupe, Lepanto, Fatima etc. Those in the lives of the Saints and Missionary Apostles)
Step 3: From Christ to Church. (in the final stage, we refute Protestants, Orthodox etc, who believe in Christ but not the Church. This part is fairly easier than the other 2 imo, because we have at least all of Sacred Scripture as a common authority).
In the same way as polytheism is disproved by natural proofs that God is One, so is Protestant poly-ecclesia-ism disproved by Scriptural Proofs (and Traditional Proofs, from Councils like Nicaea etc if they admit it) that the Church is One. For if the Church is One, and She most certainly is, it necessarily follows, that She cannot be found within the 30,000 odd sects of Protestantism; just as it necessarily follows, that the True God being One, cannot be among the many gods of any pantheon no matter how many millions they are. And thus we are able to exclude millions in one short - for the very fact that they admit other gods beside themselves proves that they are false gods - rather than refute them one by one.
Chris said: "If you are able to accept this, I would challenge you to consider the likelihood that this Prime Mover built into creation a plan of morality and religion for our temporal and eternal good, and that if you examine all the choices, Christianity (specifically Catholicism) is the religion that makes the most sense." Absolutely agreed.
Perhaps the hardest part for us Catholics, the one demanding most study from us, Chris, may be precisely that path helping someone who believes in a generic God (i.e. who is in step 1), to specifically believing in Christ as God (i.e. to come to step 2)? Would you disagree?
God Bless.