I don’t know if I’m too late but I have a few thoughts. Speaking of Burt Lancaster, one of my favorites, there are some interesting options not mentioned -
Elmer Gantry, a powerful film with great performances about a group running a evangelist revival tent con in the 1930s (sort of remade as a Steve Martin movie I never heard of I randomly came across on a streaming service that I had to turn off after a few minutes because it was so bad, but Elmer Gantry is riveting). Touches on many things relevant to your themes (though not foreign policy), and covers a different aspect than most movies covering those themes in the 20th century.
The Swimmer, a 1968 film where at first it feels like it’s a Doris Day/Rock Hudson sort of thing, but quickly becomes apparent that it’s a 60’s reaction to that (it’s based on a 1964 short story published in The New Yorker) and the catching-up of the rest of society. This is a strange but powerful film, and I think really very relevant to today too, where we’ve seeing a resurgence of the kind of upper-middle-class (and largely white) dominance like in the 1950s, leading to today’s period of civil unrest not too unlike the 60’s. It’s similar in some ways to how Mad Men deconstructed the 60’s, but it’s more abstract and allegorical (and short). The Apartment is perhaps the classic on this theme (and referenced directly in Mad Men) but I think The Swimmer would work better and be more interesting for a college class because there’s a lot to interpret and relate to today.
Judgment at Nuremberg, starring Burt Lancaster as a Nazi, is interesting and unique on the theme of foreign policy and Americans abroad, as it’s set in American-occupied Germany. Americanness being applied to a different culture is a major theme. Really lots to unpack in this one.
I understand the reasoning not to use Dr. Strangelove, though personally I would probably include it (keeping in mind that most college students today will not even have heard of it), but you may be interested in another film with a similar theme, Seven Days in May - starring Burt Lancaster, naturally. It doesn’t have quite the same impact as Dr. Strangelove to a modern audience perhaps, but it captures something unique about the time (and very differently from Fail Safe, the one that’s basically the same story as Dr. Strangelove but not a comedy) - foreign policy, but specifically how a divided and volatile American populace reacts to - or revolts against - it. Also very relevant to today.
The Treasure of the Sierra Madre has an interesting American-abroad reversal - Bogart is broke and homeless and goes to Mexico to find his fortune. Probably not relevant enough for the class, but I thought of it anyway.
At the extreme for the theme of Americans abroad, there’s Straw Dogs, about Americans who move to a small town in England and get… harassed, I’ll say. Probably not relevant enough either, but it illustrates the idea of how many people outside the US dislike Americans.