I'm not quite sure about the whole combining of dive bar and burgers in the same sentence. They don't have to overlap. A lot of the best burgers I've had are in less than dive-y surroundings. I've had extremely good burgers in sit-down restaurants, although some only had them as bar bites. I absolutely loved the Prime Burger at different Ruth's Chris locations during their happy hour in the bar area, although back then different locations might have different prices. I was on a long-ish work assignment where I went to one that charged $6, although I'd typically just get the steak sandwich. Another location was $7. I think now it's $9 (or more depending on location). That was better than any dive bar burger I've ever had.
When I was in college there was this one place I went to for a burger and fries (or onion rings) the Tuesday before my first final. And I did it for all 8 semesters. That place was really good and it's still around, although the prices are kind of high now.