OP: could the Jews of the first century, have called Mary "the daughter of Aaron", even if she was only akin to Elisabeth by the tribe of Judah by her mother's side?
Mary was of the tribe of Judah, of David's seed.
Concerning his Son Jesus Christ our Lord, which was made of the seed of David according to the flesh; Rom 1:3
The word translated by KJV as "cousin" at Luke 1:36 is "syngenes", which may also mean kin or of the same nation. Various other bible translations use it in that wider sense.
RSV And behold, your kinswoman Elizabeth in her old age has also conceived a son; and this is the sixth month with her who was called barren.
YLT and lo, Elisabeth, thy kinswoman, she also hath conceived a son in her old age, and this is the sixth month to her who was called barren;
Additionally, there is the same sense of specifically not being of Levi in Hebrews.
For it is evident that our Lord sprang out of Juda; of which tribe Moses spake nothing concerning priesthood. Heb 7:14
Further, to add, scripture calls for Messiah to be of the line of David, not Aaron.
Hath not the scripture said, That Christ cometh of the seed of David, and out of the town of Bethlehem, where David was? John 7:42
The LORD hath sworn in truth unto David; he will not turn from it; Of the fruit of thy body will I set upon thy throne. Psalm 132:11
And Jesus answered and said, while he taught in the temple, How say the scribes that Christ is the Son of David? Mk 12:35
So, to answer the OP, no, it would be impossible for Mary to be of Levi, rather than Judah.