When my mother was in her later years, she didn't drive (and had a host of other medical things). My sister and I would visit her frequently and would also take her to stay with one of us for the occasional weekend. The commute/drive was longer for my sister than for me, if that matters. My mother loved to go out for lunch or dinner during these visits; she couldn't get out on her own and really didn't like to cook anymore.
During one visit, my mother remarked to me (in an annoyed/somewhat hurt tone) that my sister always expected her (my mother) to pay whenever they went out to dinner (I can't remember if mom said that sister said it was b/c of the travel to see her, or if I inferred that). She said that my sister would hand her the check or make a remark about mother paying. But it was obvious that it bothered my mother, for her to mention it to me. After that, I just didn't worry about any perceived equality when I went out to lunch with her; I just paid. And most of the time, my mother would give me some money or would offer to pay herself. But I didn't make a big deal out of it, and I certainly never just announced to my mother that she would be paying.
I get it; my mother didn't really have any objection to paying for at least herself, but she didn't want to be looked at as just a free lunch and that was how that aspect looked, with how my sister apparently handled it. I know there was the driving involved, with the tasks and chores, but my mother didn't really see that, or didn't register it anymore b/c it had just become her normal life. But my sister could have handled it better. My mother did appreciate all that we did, but she also got crotchety at times with all of her medical ailments (it's draining).
I think this whole issue is on the cousin, frankly, for the way he handled it. If he had no intention of paying, he should have stayed the heck out of it. And really, a 50 something man telling his elderly uncle to pay for him? Seriously? But I suppose he might have had a mindset like my sister, like "I'm doing the driving, and the least he can do is pay." But like my sister, he handled it very badly, and it came across as kind of gimme-piggish and entitled.'
If your dad is still feeling weird about this whole thing, perhaps he could try a different tactic: get them a gas card. "I know I don't drive, so you always have to drive when you visit me. I do appreciate it, so I wanted to give you this to let you know that I do appreciate the effort. It's not much, but I hope it helps." Then the lunch cost is taken out of the equation. And I think the sons could start picking up the tab now and then; senior meals at most places are pretty inexpensive relative to the regular selections so them treating the two seniors wouldn't be as much of a burden as two normal meals. Of course, that's easier said than done.
If you'd like to help out your dad, how about you get him a couple of gift cards that would be just enough for his meal/beverage? Then he could introduce the concept of going dutch without seeming cheap. "Look, lkdrymom got me this gift card for XYZ restaurant! So my lunch is on her today! haha" Or even select the restaurant based on where he has the gift card, so the concept that "I'll pay for myself" is laid out on the table before they ever get to the restaurant. (He could even pay for the cards himself but just say the cards are from you, then he has the 'excuse'.)