I fail to see how it's common sense. The rule says "do this, then you can do that" for two weapon fighting (not a fighting style by the way, but a base combat rule anyone can use).
Then duelist fighting style says "if you attack with a one handed weapon and have nothing in the other hand, get +2 damage". That's it, no need to invent other rules or restrictions that don't exist in the rules.
As with everything else, it's up to the DM, who can make their own rulings, but nothing in the text implies you couldn't do this. You're certainly allowed to mix other fighting styles- if I wear armor, Defensive gives me +1 AC. It doesn't turn off with other styles.
It only "turns off" if I'm not longer wearing armor.
Similarly, if I have Thrown Weapon Fighting Style: "when you hit with a ranged attack using a thrown weapon, you gain a +2 bonus to the damage roll."
So in this instance of a guy with Two Weapon Fighting throwing two daggers, Thrown Weapon Fighting Style applies when I use a ranged attack with a thrown weapon.
The class feature only checks to see if it's valid when you use it. I would think that is common sense.