Ironically, I think I would pay 3$ more if the cheeseburger cost 3$ more than the regular burger but would say no if it was like there's only a burger on the menu and they tell me it costs 3$ more to make it a cheeseburger.
Also, from the way she tells the story, I think the "problem" in this case wasn't not getting the cheese, but the way he complained about it afterwards. Not judging whether or not it's an actual issue, but depending on how it went, it could have come off as someone being super stingy. Not digging the way she just left without telling him she was leaving (don't get me wrong, leaving is always 100% fair if you're not comfortable or don't like the person, IMO, just tell them unless you don't feel it's safe of course) but she did go to the waitress, paid for the tab and then let him know she left (however badly she did it). As far as ghosting goes, it could have been worse haha.
I don't think refusing 3$ add to get a cheeseburger is being a cheapskate. However, I think complaining about it after and making a big deal out of it kind of is. I don't know, it's not the kind of thing I would harp on to someone during our first date?