As a former Catholic (with a decade+ of Catholic schooling), I can share that that was never proposed or demonstrated.

It's an official dogma of the Catholic church that God's existence can be proved by the unaided reason. Did your Catholic education somehow omit any mention of the Five Ways?

http://www.ewtn.com/library/councils/v1.htm

>If anyone says that the one, true God, our creator and lord, cannot be known with certainty from the things that have been made, by the natural light of human reason: let him be anathema."

I am not a Christian, and I'm certainly not going to start defending Catholic arguments for God's existence here (!), but the arguments definitely exist.

Which itself is an irrational position to hold, so if that is indeed a position they hold, they hold it by faith.

No, they hold it on the basis of certain arguments for God's existence.

How do irrational arguments for the existence of a god support the position that that god's existence can be established by rational argument?

I am not taking a position on whether the arguments are good or bad. But supposing the arguments are bad, sincerely believing something on the basis of a bad argument is not the same as believing it by faith.

Well, unless that argument is built on an assumption that in turn is held to be true on faith, which makes this belief transitively also held to be true on faith.

The arguments that Catholics use typically are not based on such assumptions. See e.g. Aquinas's Five Ways, which are based, more or less, on premises from Aristotelean metaphysics.