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.