It sounds like you are not only looking for a rational argument for the Authority of the Catholic Church, but a rational argument for Christianity as well. For this, in addition to a rational argument for the authority of the Church, please sit tight and read my long post: (By the way this is not how I personally came to believe in the Church, but it is a very rational argument which supports my belief)
1) The New Testament is the most bibliographically sound ancient written work.
Take a look at this chart: carm.org/manuscript-evidence
How often do people question whether or not Plato actually conveyed the ideas we have in our current copies and translations of his works? Almost never. If the message of these other ancient teachers aren’t questioned, why should Jesus be any different?
2) In the New Testament, Jesus claims to be God
But he held his peace, and answered nothing. Again the high priest asked him, and said to him: Art thou the Christ the Son of the blessed God?  And Jesus said to him: I am. And you shall see the Son of man sitting on the right hand of the power of God, and coming with the clouds of heaven.
(Mark 14:61 Douay-Rheims version)
Jesus said to them: Amen, amen I say to you, before Abraham was made, I am.
(John 8:58 Douay-Rheims version). Note that ‘I am’ is in present tense, in reference to the name of God given to Moses at the burning bush. (Exodus 3:14) The One Who Is, the Being Who IS existence itself. To God, there is no past or future since He is eternal and outside of time, hence the present tense. He does not use the natural grammar construction “before…I was”. He uses “before… I am”.
There are many other places where he claims to be God as well.
in John 20:27-29, Thomas says that Jesus is God and Jesus acknowledges that he has “believed” and does not reproach him for it. John 14:1-12 is another one.
3) If Jesus claimed to be God then He could only be either…(C.S. Lewis’ “Liar Lunatic Lord” argument)
**a liar **(He wasn’t God and knew He wasn’t),
**a lunatic **(He wasn’t God but He thought He was),
or The Lord (He was God).
There is evidence that he was neither a liar (would a liar go to the cross for his lie?) nor a lunatic. He must be God. Believing that Jesus was just a “kind man”, as many non-Christians do, is clearly not a supportable position.
So far, this gets us to Christianity in general. Now for Catholicism in particular:
4) In the New Teestament, Jesus founded the Church in history.
The verses mentioned by the other poster supports this. (supports it for historical accuracy, i.e. Jesus actually said those words, and the early Church actually practiced those things) posted by JRKH:
A) Scripture clearly demonstrates that such authority exists - given by Jesus in both Mt 16 and Mt 18 - and exercised by the early Church, as recorded in Acts 15.
Mt 16:19 says:
And I will give to thee the keys of the kingdom of heaven. And whatsoever thou shalt bind upon earth, it shall be bound also in heaven: and whatsoever thou shalt loose upon earth, it shall be loosed also in heaven.
5) If Jesus was God, and He said what is recorded in Mt 16:19, then that means God gave the Church authority.
Note that this argument does not require one to believe in the infallible authority of scripture before hand, only in the historical accuracy of it.