As the title indicates, how can I set up my homebrew options so they can appear alongside Defense, Dueling, Great Weapon Fighting, etc.? If it matters, these fighting styles won't be shared; they're specifically for an adventure I'm building.

There is no way to alter base class features like Fighting Style at this time. You will have to create custom versions of every subclass for all three classes, and then put in a feature that they could use with a note that it replaces standard fighting styles. 

Or you could make it a Feat, and just tell your players if they want it not to choose a regular fighting style and you could place the feat directly on their character sheet without them needing to spend an ASI.

Quote from IamSposta >>

Or you could make it a Feat, and just tell your players if they want it not to choose a regular fighting style and you could place the feat directly on their character sheet without them needing to spend an ASI.

Thanks. I'll follow the Feat recommendation. With that, a character sheet's build won't go wonky when the Fighting Style selection is ignored, will it? I wonder whether an unassigned fighting style will mess something down the line.

Quote from Miromonti >>

Quote from IamSposta >>

Or you could make it a Feat, and just tell your players if they want it not to choose a regular fighting style and you could place the feat directly on their character sheet without them needing to spend an ASI.

Thanks. I'll follow the Feat recommendation. With that, a character sheet's build won't go wonky when the Fighting Style selection is ignored, will it? I wonder whether an unassigned fighting style will mess something down the line.

 
It'll be fine.