As you may be aware, getting to that budget as an international is the trickiest bit as need aid for internationals is relatively limited and the competition for it is fierce, and $40K is below the full cost of attendance for international students at many public universities (which usually do not have any need aid for internationals).
A few suggestions to consider:
St Olaf is great for Classics and specifically Latin or Latin-based Classics. They have a robust need+merit program which might get you within budget:
https://wp.stolaf.edu/financialaid/international-student-information/
https://wp.stolaf.edu/classics/
Gustavus Adolphus also has a robust Classics program, including a Languages major with a Latin concentration option, and they have a specific combined merit/need program for internationals that could get you in budget (the Count Folke Bernadotte Scholarship):
Oberlin is generous with international financial aid if you can get it–more competitive than domestic but you strike me as someone they might really see as a great fit. People sometimes think of them in relation to the Conservatory but they are ALSO a top academic LAC and have a robust Classics program with a possible Latin major:
I’m just going to rapid fire some more names of broadly similar colleges which I know do Classics–you can look up their international aid for students pages and Classics department pages like the above to see if they seem like good opportunities. That list would include Wabash, Dickinson, Skidmore, Rhodes, Kenyon, Grinnell, Carleton, Macalester, and Ohio Wesleyan.
Another idea would be to look at women’s colleges. Bryn Mawr has a legendary Classics program which then extends out through its consortium relationships with Haverford, Swarthmore, and Penn. I know it has a lot of international students, I am less sure what its aid looks like, but could be worth taking a shot:
There are then also a bunch of very highly selective private universities with good aid for internationals if you can get admitted and with excellent Classics departments. Brown, for example, sounds like a great fit for you. Everyone knows Penn for Wharton, but their CAS is fantastic and very strong in Classics. WUSTL, Georgetown, or Emory. But you need to check if these sorts of colleges would be within budget for you, and they can’t be Plan A, just possible “reach” applications that would be nice if they worked out.