When a champion fighter, with "Two Weapon Fighting" fighting style and the "Dual Wielder" feat, picks up "Dueling" for his 2nd fighting style... Is he able to make use of the +2 to damage rolls with a single weapon for the first hit and only draws the 2nd weapon later to deal the 2nd hit in his extra attacks (in order to trigger the requirement for twf) to deliver another attack in his bonus action?
The key point is in the description of two-weapon fighting:
Two-Weapon Fighting (PHB Pg. 195)
When you take the Attack action and attack with a light melee weapon that you’re holding in one hand, you can use a bonus action to attack with a different light melee weapon that you’re holding in the other hand.
Question is, when you take the Attack action, is there a weapon in your other hand? If there is, you can't get the benefit of Dueling. If there is not, you can't make an off-hand attack.
(Note the present tense "that you're holding in the other hand", not "that you will be holding later in the turn" or "that you were holding earlier".)