King of the Storm will not respawn once you've killed him and the fight will start directly against Nameless King.
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This mod make it so that King of the Storm doesn't respawn once killed since it doesn't make much sense to me that he comes back from the dead every time and the fight itself is little more than a time waster compared to Nameless. If you start the fight again after his death the fight will automatically jump to the Nameless King phase.
This is more of a proof of concept that I made just to see if I could than anything serious and therefore it's not thoroughly tested. This may or may not get you banned and may or may not break your save so do play offline/family-shared and back up your save file if you want to try it out. I don't think either should happen but I don't really know what I'm doing so you never know.
INSTALLATION:
You need to use UXM to unpack the game files first and foremost, and don't forget to click patch.
Once that's done go to the DARK SOULS III/Game folder and extract the event folder in my archive there, it should ask you to overwrite m32_00_00_00.emevd.dcx. Done, you can go fight Nameless now.