When counting the Ushpizin chronologically, Moshe and Aharon are the only ones who lived at the same time. In fact, Moshe was younger, so it would make sense to put him second (as you asked). A couple possible answers:
In Shemos 6:26-27, Rashi points out that the reason one passuk says Aharon's name first and the other says Moshe's name first is to show they were equal. However, a lot of the Commentaries on Rashi explain Moshe was greater than Aharon, but in this case they were equal. See here, here and here.
The Ushpizin are referred to as the "shepherds of the jewish people". Moshe role as shepherd was unarguably greater than Aharon's role.
The Jewish Press, in a Q&A that discusses Ushpizin, says "we find Moses always preceding Aaron, even though Aaron was his elder."
Ta'ami Minhagim (also brought in the Jewish Press article) give a reason for the Kabalistic order:
- Aharon passed away on Rosh Chodesh Av. Rosh Chodesh Av always falls out on the same day of the week as the 5th day of Sukkot. Moshe desired the same death as Aharon, and Hashem granted him his wish. Therefore the 4th day of Sukkot is always the same day of the week as the previous Lag Ba'omer, the passing of Rashbi. Rabbi Shimon Bar Yochai was a Gilgul of Moshe Rabbeinu. (Incidentally, one of the reasons given for why the passing of Rashbi is a holiday, but the passing of Moshe is a fast day, is that we fast because the soul of Moshe Rabbeinu never entered the land of Israel, but it did when reincarnated as Rashbi, and therefore we celebrate on the day of his passing)
The Taamei Haminhagim, quoting the "Kocho D'Rashbi", points out that there is one place in the Zohar where the order of the Ushpizin is mentioned chronologically. He then connects this to a hint for the order of the Ushpizin in Parshat Vayechi, that could be interpreted either way.
The Pasuk in Bereshit 48:3 says:
וַיֹּאמֶר יַעֲקֹב אֶל יוֹסֵף אֵל שַׁדַּי נִרְאָה אֵלַי בְּלוּז בְּאֶרֶץ כְּנָעַן וַיְבָרֶךְ אֹתִי:
So, according to the Chronological order, the Passuk has 1) Yaakov, 2) Yosef, 3) Moshe, 4) Aharon.
According to the Kabbalistic order, Yaakov is Telling Yosef that he wants Moshe and Aharon to be "Eilay", next to him, i.e. preceding Yosef in the order of Ushpizin.
I haven't seen the Sheloh inside (a link would be nice), so I don't know if he gives a reason for his order, but according to all these reasons we can see why Moshe comes before Aharon.