[Burgers]

I've spent a lifetime in the food business so I know a thing or two about food safety. The odds of you getting food poisoning and having reactions in a 3 hour flight from MIA to JFK are virtually nil. Don't take my word for it, look it up, you'll find latency for food poisoning is not measured in hours but days more often than not. Its extremely rare for it to be fewer than 8 hours - again do your own research if you doubt me. The nastiest enterics will get you that sick but airline caterers follow extremely strict handling guidelines - I've toured a few flight kitchens in my time.
BA is not going to advocate for you as they are two separate entities that share a codeshare and alliance relationship for the purpose of sales and marketing, not liability. American is not going to compensate you because you have no proof that they got you sick, besides when you seek acknowledgement it implies culpability add to that you mentioned the word "sue" it looks like you are seeking damages, no company in their right mind will talk to you at that point. In other words now you don't exist.
Food poisoning sucks but it happens - humans are fragile creatures and simply touching a bathroom door can get you that ill after your next meal.