Can you benefit from the Dueling fighting style after having thrown a light weapon?

So, the PHB has this to say about unsheathing a weapon (page 190)

You can also interact with one object or feature of the environment for free, during either your move or your action. For example […] you could draw your weapon as part of the same action you use to attack.

So, if I had already a short-sword in one hand, I could technically draw the dagger dangling at my belt and throwing it as part as the same action, correct?

And since they’re both light weapons, I could then use my bonus action to attack with my short-sword (without the proficiency bonus on the attack roll), right?

But would that attack benefit from the +2 damage bonus from the Dueling fighting-style, since I did attack with only one weapon in hand, or wouldn’t it count since I technically used another weapon during this turn?