So here is what I have for a Scholar of the First Sin Any% Route. I have not run this route yet, this is all theory. The majority of this is still the Any% Current Patch route from Dark Souls 2 vanilla, in fact once you get to the 2nd Rotten fight, the runs are again identical with exception of one item pickup before Velstadt.
The obstacles that SotFS presents that were not an issue in DaS2 are:
You now require 4 Fragrant Branches instead of 1 due to increased amount of petrified NPCs blocking the way in Lost Bastille, Dragon Aerie and Black Gulch.
Going into the Sunken King DLC is imo no longer effective in this run. To enter the DLC you now need to collect the Dragon Talon from the Forgotten Door down the Mejula well. To open that door you need the Forgotten Key obtained from killing one of the giants underneath Black Gulch. I don't believe that's feasible in the run, especially considering how early you are expected to kill giants. A result of not entering the DLC means you are at a 22,000 soul deficit off the 1 million necessary to enter the Shrine of Winter. You are also short one of the necessary Bonfire Acetics and will never acquire Flynn's Ring - so reduced damage.
So the changes I made from the Any% Current Patch DaS2 Vanilla run are:
Pick up extra branches - 2 in The Gutter, 1 in The Lost Bastille and 1 before Velstadt.
Kill Najka and The Congregation. This will make up the soul deficit. In fact, after killing these two bosses and using their souls you end up at about a 14,000 soul surplus. There is also a Bonfire Acetic above The Congregation fight.
Removed a couple soul pickups from the vanilla run to use up some of our soul surplus.
Slightly altered the route once you get to the Forked Road to accompany our trip to Najka/Congregation.
We now purchase an extra Gold Pine Resin from Vengarl for use on Najka.
Alright so here's the run below. Would love any feedback/criticisms/suggestions. I am certain this is not even close to optimized.