I would say, rather, that no argument from reason will be convincing if not made with love. But if your heart and life are conformed to Christ, you might suggest one of the Ten Prayers God Always Says Yes To, in this case, "God, Prove to me that you exist."
Here is how I think it ought to be set up: "You're a person of reason, right? So among other things, you have to admit that you don't know everything and could be wrong, and you want to know the truth, no matter what it is, right? You've asked me to prove God's existence to you, and you're right, that I can't. But God can.
"Well, let's presume that the God that the Catholic Church proclaims is real. That is, He loves us all without limit or exception, knows everything, can do whatever He wants, and made us to know Him as well as we are able, and love Him as much as we can. Now, if all of those things are true, you can test for His existence.
"Silently, within the vaults of your mind, say to Him, 'God, I've heard the claims that the Catholic Church has made about you, and I don't believe it. It's utterly ridiculous. Prove to me that I am wrong.'
"Now, if He does not do this for an honest seeker of the truth, it means one or more of the following things:
- He doesn't care.
- He doesn't know.
- He can't prove it.
- He doesn't exist.
- Or, the seeker doesn't really want Him to prove His existence to you.
If this test would be convincing, it would be from God's work, and not my own.