「そりゃあ持{も}っているでしょうに。」
For the majority of native speakers, this is not a regular, "calm/relaxed" way to say:
"(Yes,) I have (a) credit card(s)."
That is why I asked above how exactly you asked your question that triggered the use of the highly nuanced and possibly emotional に-ending.
My first impression when I read this question before the edit was that the speaker was slightly irritated or surprised at your question -- "Of course, I have a credit card. Why would you even ask?". That could happen because credit cards are far more common in Japan than what the popular J-vloggers on YouTube seem to want you to believe.
According to goo辞書, 「(よ)うに」 does the following:
★後悔・不満・非難・疑問・感動などを表わす。
That means "expressing regret, dissatisfaction, criticism, doubt, emotion, etc" and this is precisely why I said what I said at the beginning. Your friend's reply is clearly nuanced. A regular, non-nuanced reply would have been along the lines of:
「(ええ、)持ってますよ。」
The difference between that and the actual reply you received is huge.
What one could still not tell is exactly why s/he replied that way. To analyze it, we might need to read the entire conversation that took place, which I will not ask you to show us. As a native speaker, I could just almost hear the silent 「なんでそんな(当たり前の)こと聞くの?」.
I asked my friend if they had a credit card← What were the exact words you used to ask that? IMHO, that would be the key for a productive answer. The reply sounds fairly nuanced; therefore, a good amount of context in the original Japanese would help a ton. A long explanation in English would not. – Nov 11 '19 at 11:44