I managed to get this run today, managed to get everything done and boss fight enterable at round 13, earlier is definitely possible, I'm thinking with perfect luck round 6 or 7 would be possible (although you would probably be underprepared to actually take on the boss)
Basically to do this ensure that you get your first Klaus arm during the first special round, Ensure as much is done as possible. the important parts for this are: placing the brain in the computer in the teleporter room, having the first part of the Gorgofex from the lab and I'm not sure on this but you may need to input the 4 digit code into the computer in the teleporter room as well.
After doing all of this you can begin the uberklaus/Forsaken fight by surging using the 4 computers. However the first thing you should do once the normal Uberklaus spawns is take it down the elevator and activate Klaus (don't worry, you will get the other arm soon, just electrocute Klaus for now). Then take the UberKlaus back to the forsaken room and do this as normal, but during the part of the fight were Klaus' spawn in this is where you will get the other arm, shoot the arms off of them as you would normally do, this dropped successfully for me. Then proceed to kill the forsaken, craft the gorgofex and go visit friendly Klaus to activate his little mission. From here proceed to do the EE as normal.
By utilising a few side easter eggs (maybe getting lucky with the recycle bin aether tool or vending maching) or gobbles I think the boss fight could be doable as early as round 6 or 7 depending on if any round flips are necessary for the EE
I have 1 theory that may allow you to do it earlier. Perhaps by failing the forsaken step by killing the uberklaus (and then the core) without allowing the forsaken to spawn you might get lucky with a klaus arm, From here I'm unsure if you have to flip the round or if another spawns in automatically, I'm also unsure if the cores can even drop the arms to begin with. If anyone tries this do let me know