For the ice cream: Pour the cream into a large chilled bowl and whisk to firm peaks, either by hand or with an electric mixer. Pour the sweetened condensed milk into a second large bowl and mix in the vanilla and passion fruit nectar. In three additions, fold the whipped cream into the condensed milk until just combined. Pour the mixture into an 8 1/2-by-4 1/2-inch loaf pan and cover with plastic wrap, pressing lightly on the surface to keep air out. Freeze until firm, at least 4 hours and up to overnight.
For the caramel sauce: Pour the sugar into a small saucepan with 1/4 cup water and set over medium heat. Add the lemon juice. Scrape in the seeds from the vanilla bean halves and add the pod. Bring the mixture to a simmer and cook, swirling the pan occasionally and brushing down the sides of the pot with a pastry brush as needed, until the sugar syrup is a dark amber color, about 10 minutes.
Carefully add the cream--it may bubble up--and stir to combine. Whisk in the butter, a few pieces at a time. Add a pinch of salt, then whisk in the rum. Let the sauce cool until just warm, about 15 minutes.
While the sauce is cooling, remove the ice cream from the freezer to soften. Serve the warm sauce over the ice cream.
Store any extra caramel sauce in a sealed glass container and refrigerate for up to 1 week. Rewarm before serving.