If I wanted to travel from Voyageurs National Park into the BWCA at Little Vermillion Lake (River?), is there a way to do so in a timely way? Essentially, is there any way to avoid having to leave the area to pick up a BWCA permit before continuing towards LLC?

Sure

Pickup your entry permit (Little Vermillion) at an outfitter in ely when they open (around 6 am for most). Depending on how hard you want to paddle and where within Voyageurs you start, either choose that day for the entry date or the following day. Drive up to Voyageurs and launch as early as 9 am +/- and plan your length of day paddling according to when you need to be at Little Vermillion Entry Point (either the same day by midnight or the following day at midnight).

Permit requirements are that you must enter the day of or the next day of permit issueance. This could give you 2 days to reach the edge of the BWCA.

I corresponded with the Superior National Forest Supervisors Office about this a few years ago. I am pretty sure this would still apply. I was planning to start at the western end of Rainy Lake and spend several days traveling through the park before my entry date. They said they could accommodate me. I doubt that I will still have that email, but will look. Call or email the Supervisors Office. The response I got was from the Forest Supervisor herself.