Our good buddy Ken has nailed most/all of the points at this point, but just wanted to drop in and let you know I got your message and headed over here to answer them. Ken beat me to it.
If you happen to be on facebook we have a small but growing facebook group called Lost Coast Trail that you can join and yap with other LCT hikers about a lot of this, as well as stay up to date with issues, laws, weather, trail condistions, and such.
A thru-hike of the entire LCT (North & South) is fairly easy to plan, with the exception of transportation. I really have no idea what the two "licensed shuttle companies" charge for pick-up at the Southern terminus of the Southern half, or even they even offer that. Guessing 99% of what they offer is either at Mattole or Shelter Cove.
As Ken has stated, the Southern half of the LCT is soooo different from the Northern half. Gone is the beach hiking, lots of easy access to water, wonderful fog and long views of the beach and cliffs - replaced with massive elevation, limited water access, much hotter weather conditions, and a lot more remote access.
That said, in many ways I prefer the Southern half, but it is not really the iconic LCT that the Northern half is. The massive crowds of the Northern half keep me away from there. I have hiked most of the Southern half (and a few parts multiple times) and have never seen a single person on the trail.
Getting from black sands beach to the start of the trail of the Southern half is not all that hard logistically. Just involves some road climbing out of Shelter Cove.
Ken brings up an good point about the bear canister issue. The first day of the Southern half you will be legally required to carry a bear canister. The other two or three days you will not. No viable way to get it shipped out so you just have to suck it up and carry it the entire way, but that is probably a good thing.
Without a doubt the cost of the BLM authorized shuttle companies is a big contributor to why a lot of people just yo-yo the Northern section of the LCT. It is 98 mile, 4 hour, round trip drive for them (the SC->Mattole option). For them to drive from SC to Usal road (given they use vans, probably no way they will meet you at the beach, forcing you to dirt walk over to hw 1, of boy the fun) is ~50 miles, they they have to drive the ~90 miles up to mattole to drop you off, and the 50 miles back to SC (which is where I am guessing they live.) So all that works out to about 190 miles and about 6 hours of driving time. So basically a tank of gas and an entire day spent on the road. After waiting around, loading up, cleaning up afterwards, and such, probably works out to a good 7 or 8 hours of work on their part for the full Usal to Mattole shuttle. I did it once for a buddy... Eureka -> Usal -> Mattole -> Eureka and it was literally an entire day of driving... and I only drove about 300 miles... which was an entire tank of gas for me. I love to drive, but that is a loooong time driving for such a short amount of miles.
If time is not an issue, give the entire LCT a yo-yo! Very very very few people have ever done that. Start off with a few days of food. Resupply in SC. Once you get to Usal (Mattole if you go nobo), turn around and head back to SC and resupply again. Would be an awesome way to spend a week out on the trail!