Whisk the pomegranate juice and lavender in a chilled stainless-steel bowl. Add the lime juice and rum, if desired. Freeze until icy around the edges, 20 to 25 minutes. Whisk, then continue to freeze, whisking every 20 to 25 minutes, until slushy, about 2 hours, 30 minutes; cover and freeze overnight.
Scrape the granita with a fork, then scoop into martini glasses. Garnish with lavender.