It wasn’t post-apocalyptic, but the rest of your description – especially withn the “performing surgery” – fits to a T the film Killers from Space. I watched it frequently on TV in the early 1960s, and now have it both on VHS and DVD. I;ve shown it at my Bad Film Festival, where it fits in perfectly. MST3LK didn’t do it, but they covered it in the short-lived Film Crew series.
This shows some shots from it, including the surgery scene:
http://www.moviemonstermuseum.com/kfs.html
I see Yevgeny has already mentioned it, with a link to the Wikipedia page.
I can’t believe that this isn’t the film – the bulging eyes and the surgery make it certain, in \my book. And the aliens were driven underground – on their home planet – and developed those big eyes. A young Peter Graves stars as Doug Martin.
The eyes always looked comically absurd (although scary to a 5 year old). They never did get the lined up properly, so they’re always a bit walleyed or crosseyed.
The Videotape Horrible Horror actually contains out-takes from this film – shots of them trying to get the sluggish lizards to react for the camera.
My favorite part is the end, where the filmed-from-a plane shot of the atomic bomb going off is passed off as one seen through a window. As9ide from the perspective being completely wrong (and they’re looking at an ocean, no less), it’s hilarious because you know that an A-bomb has gone off, because the venetian blinds rattle. Low-budget film making at its cheesiest.
I’m downloading “Killers from Space,” since it seems the likeliest. I was 8 or 9 when it came out, and that seems about right. Maybe if I watch the whole thing something will jog my memory. Something I should have mentioned before but it didn’t occur to me: I’m positive the film was in black-and-white.