While I'm certainly not trying to be quas-religious, what's the point of a definition of it means nothing? If there's meat in the patty because of poor preparation, that's plainly easily avoidable by scraping the grill, and bk just refuses to do it or find one of the other amazingly easy ways to not have it cooked in beef fat, you can't call it vegetarian. Absolutely there's no way for a company to never have insects, rats, and other pests in their foods even under the best of conditions. This isn't that. If they cook the patty in beef fat, is not meatless, you've made the beef fat an intentional part of the product, therefore not vegetarian. The patty itself is still meatless, the end product is not.

Edit: Most vegetarians you happen to know. I personally don't like the fluidity with which the term is used expressly because it leads to confusion. You want to eat meat, cool you do you, but don't make it harder to explain for people who choose to actually follow the definition by making it meaningless. It's like the term "green" when people guest started using it it had a definition that was understood, now it's meaningless. If someone is comfortable with meat intentionally being in their food for to preparation to the point that they don't mind if there's meat in their food, they probably consume far more meat products than they realize and I doubt that they are as vegetarian a they intend to be, which again is fine, I just dislike claiming to be something that, by definition, one isn't and trying to torture the definition to your preferences.