What's the minimum time needed to traverse BWCA completely from West to East?

Paging Beav to the courtesy phone..

Depends whether you want to paddle in the dark or not.

First light to dark would be doable. Obviously all is weather-permitting.

Well...the fastest known and documented time is 1 day 22 hours 40 minutes. That's from the western entry point off of Sandpoint Lake to the eastern exit point of North Fowl Lake.

Night paddling required for that.

BeaV of course is right if you are asking what is the fastest time this route has been accomplished.
If you are asking how long a mere mortal person can do this trip I can give you some information.
From entry point to entry point that BeaV describes via the border route is just about 170 miles. But you’d have to start at Crane lake and end at John Lake entry which will add about 10 miles for a total of 180.
Paddling in June you can get about 16 hours of daylight and average about 3mph counting paddle and portage, some headwind, some tail wind. To actually average 3mph you need to be cruising at 4mph because of breaks and portages and so forth. There are a few really difficult areas to navigate, nearly everyone gets turned around some and wastes time. This technically gets you 48 miles per day but that is really hard to do. 40 miles average is more realistic over the course of the trip and even that requires a substantial effort.
So, 180 miles at 40 miles per day is about 4.5 days.
If you are going to do this trip you really need to add the area outside the BWCA and end up at Grand Portage Fort.

Yes, Crane (E.P. #12 Little Vermillion) to Grand Portage Fort. The "Grand" is a 9 mile hike, so plan accordingly. Most of the Minnesota Border Route Challenge paddlers do this route in less than 8 days. There are 70+ year old paddlers doing it in 5-7 days. Check out the Water Tribe site - watertribe.com for the stats.