I’m also in the “invite them to dinner” camp. I can see not attending the wedding if you think your sister would be uncomfortable (or maybe going to the wedding itself, but not the reception, if they are in different places). I would definitely, definitely send a nice gift.
I have a large extended family, and there is always some kind of beef going on between someone and someone else. Part of maintaining a large family is working around those situations, keeping the scope of the dispute to a minimum, making certain there are lots of potential avenues of communication, and always leaving the door open to some sort of reconciliation in the future.
My wife has been estranged from one of her sisters periodically over the last 40 years. Different temperaments, different life styles and choices, different “sides” in their parents’ divorce, money disputes, and a childless woman’s discomfort at spending time with a family whose life revolved around young children. Nevertheless, as our children reached adulthood, they developed their own relationships with this aunt, which have held up remarkably well even when the sisters weren’t speaking to one another.
It has been healthy for the sisters’ relationship, too. Even when they weren’t talking, the sister could rant to our daughter about whatever was bothering her, and know perfectly well that the message would get through, and at the same time our daughter could say, “You know, you are making me uncomfortable, and I think you are making some wrong assumptions about my mom,” and that would get heard, too.