Oh my daughter loved her binkie. We only used it for bedtime and naps. But she did not want to give it up. Bridget's idea sounds great - try getting her to choose her own nighttime soother - then she will be the Big Girl in charge of herself and feel in control of giving up her binkie.
I wouldn't have the heart to have Santa be the bad guy

Even if he was taking it for the babies. It would feel like he was stealing.
FYI: fathers aren't always on the same page as mothers with the weekend visits. They often take the path of least resistance and give in way more than they should. Do not send a binkie when they go to daddy's house.