While we all try not to offend people, they are offending us. I would be upset if someone came to a restaurant and did not order a thing but pinched food from everyone else while eating the free bread. That is so tacky, yet, the social norms put people like you in this situation on the defensive when you were the one who did nothing wrong. What a shame that you even have to deal with this situation.
Either your cousin has money and refuses to spend it or she just has no discretionary money to spend at all. Either way, I would no longer invite her out to eat because she clearly is not up for the idea of dining out, so why bother asking her?? If she balks that you stopped inviting her, you can simply say "Well, you never order any food, so there really is no point in inviting you out to eat."
Instead, do spend time with her. Perhaps everyone else is not interested, but you sure can go to her house and watch a movie every so often, can't you? You can arrange a picnic on the town green, by the lake or in a state park where everyone brings their own bagged lunch for a nice peaceful outing. Invite her to local festivals or town events that have no admission fees. This way you can spend time with her without spending money. You know, spend some one on one time with her if your other friends are not up for free bee socializing.
I will add: I was once invited out to eat with four other friends, one of which brought a stranger who we were all willing to meet. We arrived at the restaurant and received our menus and water. The friend showed up with her friend and once the other friend saw the menu, she said, there is $29.99 items on the menu, I can't eat here, the prices are too high. Someone else said, that is a dinner lobster, we are here for lunch and there are options under $10.00. But, noooo, she refused to eat there, so we embarrassingly left. I snuck a $5.00 onto the table because the poor waitress had to clear all of our glasses and reset the table. So, we go down the street to a Mexican restaurant. The friend's guest ordered a $4.75 tossed salad...damn it, she could have ordered that at the other restaurant!! Anyway, during the entire meal, she was foraging food from everyone elses plate. Mind you, we were all strangers to her. She scoffed up all the salsa, sour cream and guacamole and chips, then had the audacity to take our uneaten food and add it to her salad while we all sat there watching her eat after we were all finished. I told my friend that I never wanted to dine out with that person ever again. It was awful and terribly embarrassing.