If I care more about the difference between a $3-5 meal at Burger King and a $10-15 meal at a neighborhood Italian restaurant than someone else, it's not unreasonable for me to figure "I want their company, and I want the experience of Cucina Italia rather than BK, so I'll pay for both our meals, my budget has room for that," even if the other person would be as happy eating at Burger King. Some people have stronger food and restaurant preferences than others, and neither person is wrong in that context.
That said, "Cucina Italia isn't in our budget, can we go to Burger King or somewhere similar after we play the games" is an appropriate way to put it, since OP said she wouldn't mind the Italian place, it's just not in her budget. (If she disliked Italian food, that would be a different conversation--"Oh, aunt, we'd live to see you for the day, and the games sound good, but you know I don't care for Italian food, how about Burger King instead?" with room for the aunt to suggest a Greek diner or other non-Italian choice if she disliked BK for whatever reason.)