What is the best brand of sweet vermouth for making this wintery Negroni?

I would stick with one of the old school Italian producers for this usage, either Cinzano or Carpano.

While Europeans enjoy bitter flavors, most Americans do not. I love Carpano's Punt e Mes (a vermouth-like aperitif that has some bitterness); I usually have it on the rocks with a splash of soda and an orange flag (much like a Campari soda). If that family member enjoys bitter flavors, you might consider picking up a bottle of this.

Haven't tried all the ones in these two recent review articles.
See if your local store either has these or has another recommendation.
Last, some of them are already so loaded with herbals that I wonder if adding more spices and mulling them, even before mixing the Negronis, is a good idea.
Maybe try a small sample before your guests arrive.
https://www.thespruceeats.com/best-sweet-vermouth-5090115
https://bespokeunit.com/wine/vermouth/