So, thinking about it, I decided that we could probably get the job done with a fighting stlye, and we could probably get both the fencer and the versatile specialist in one go. Also, ideally it would be mechanically unique and not just a re-skin with identical stats to another option. Here is my take on it:
Looking at the fighter, I think we can assume a conventional floor of 4.5 weapon damage per attack and X AC (the AC bonuses are all flat, so we don't need to be specific). Here is how the relevant fighting styles affect that baseline:
Defense: Either +3 to AC
or +1 to AC and +2.5 damage depending on if you take a shield or a great weapon.
Dueling: +2 to AC and +2 to damage (because you get a shield).
Great Weapon Fighting: + 3.83 to damage (because math).
So I think this is a reasonable addition:
New Fighting Style said:
Single-Weapon Fighting
When you wield a single weapon without the two-handed property and no shield, you gain a +1 bonus to attacks rolls and a +1 bonus to AC.
Adding it to our chart, it looks like this.
Defense: Either +3 to AC
or +1 to AC and +2.5 damage depending on if you take a shield or a great weapon.
Dueling: +2 to AC and +2 to damage.
Great Weapon Fighting: + 3.83 to damage
Single-Weapon Fighting: +1 to damage, +1 to attacks, +1 to AC.
That assumes a versatile weapon, with a rapier instead you don't get the +1 to damage, but you do get to stack your AC, attacks and damage all onto one stat which I consider an advantage. Also, it provides the only theoretical option for the champions additional fighting style at level 10 to directly stack, which is taking duelist on top of single weapon fighting and using a rapier. I'm not sure how I feel about that, I think it is ok, though. If your DM doesn't let you use one hand to cast spells while wielding a two handed weapon and he somehow, miraculously, lets you use this, it would be a buff to weapon-wielding casters.
Other than that, I think it fills both of my needs and is mechanically unique, so it fits the bill. What do you guys think?