I wonder how many times your dad has expected his sister to pay for lunch when they get together. If it's to the point where the cousin is pointedly saying, "Mom bought last time, you ante up this time," then maybe your dad has been taking advantage.
Sure, Aunt is the one suggesting the get-together, but I think most people expect restaurant meals to be dutch--most of the time, anyway.
And if your elderly father never travels to see his sister, but she is the one who always does the traveling, then he shouldn't be expecting her to be treating him all the time. It's not like it's a special occasion.
At the *very* least, I think your dad should have been saying, "How much is my share?" instead of assuming someone else would buy his lunch. Because, as I said, I think that unless a specific invitation has been issued as a treat, the default for restaurant meals is "everybody pays their own."
I don't agree that "reaching for the check" means "I intend to pay." I think someone has to pick up the check, and that being the one to do that is not a commitment to foot the bill for the entire party.