You are actually making several points, though I don't think you realize it. They are:
- How a child is parented (inconsistently, poorly, etc.), decisions about their healthcare, and how they are manipulated can affect their behavior. Nobody disagrees with you on this. You are absolutely correct.
- If you cannot counteract those negatives by the other parent, then "you cannot treat them as your own." This, I disagree with wholeheartedly.
- If you can't control their behavior then you must exclude them. This, could be your only course of action in certain circumstances, but if you use it for everything, and especially as the OP is suggesting with family events, then, in the long-run you will cause more harm than good. If the go-to response is to exclude the child, then it is an excuse for your own convenience and ease.
The part that I take issue with is #2 . . . and using #3 in either all situations, or the important ones for your own convenience. When a child feels as though they are not loved, not included, and the parents prefer the other children, this becomes a part of them, affecting their self-esteem, their ability to trust, future relationships, and how they turn out in the end.
Nobody ever said parenting was easy, and certainly blended families are more complex and difficult to manage. But you, as the adult, signed up for it, and are obligated, and should understand this, that you show the same empathy, compassion, love, care and concern . . . the same interest and inclusiveness to a step-child as you would your own, no matter their behavior. The key word here is "child." They do not have the same level of brain development, nor capability to reason and understand the cause/effect relationship. They are learning from successes and failures, and doing so in a poor environment half of the time.
Because of the latter, you are certainly entitled to frustration, but you do not give up on the child. My recommendation would be to work through legal systems to at least get counseling in place . . . . independent counseling for the child, and counseling with parents (all of you) separately.
When you start making "good behavior" a condition of inclusiveness with a step-child, you are setting yourself up for life-long issues as the odds of your own biological children always acting perfectly around the step-child are slim. When the step-child sees that you exhibit more tolerance, love, care and concern for your biological children when they act out or act up, this will cement her suspicions that you love them more, which creates trust and self-esteem issues, and will cause poor family dynamics and relations for the rest of your life.