First time poster, long time gamer.

So my 6th level Fighter just picked up Two Weapon Fighting Style from TCoE.  Can the character use the bonus action from Two Weapon Fighting to throw an additional dagger? Additionally, acknowledging that the off hand will not get the strength bonus to damage, would it still get the Dueling Fighting Style Bonus? 

I guess it comes down to where in the throwing do you count it as wielding two hands and not, as the Thrown Weapon Fighting Style makes the attack a seamless part of the attack.  There are Light melee weapons in both of the character's hands turning this turn; where within all of this does it switch from one handed Dueling to Two Weapon Fighting, if at all? Can you stop in the middle of the main hand draw to draw and throw from the off hand, before completing the throw with the main hand? Do you have to start with a dagger in your hand before the turn begins for it to count?

Answer as detailed as you'd like; feel free go all out.

Can you clarify which fighting styles you have; Two-Weapon Fighting, Dueling, Throwing? And are you intending all attacks to be made by throwing daggers?

Note also that this question will likely be disrupted by the still unsettled questions of whether you can make the TWF bonus attack between your normal attacks or only after all normal attacks, and whether to qualify for TWF you must actually be holding both weapons at the same time during one of your normal attacks or if you can draw the second later. (Personally I allow the bonus attack to happen in any order, but only if at least one attack is made while holding both weapons involved.)

Just the Dueling & Thrown Weapon Fighting Styles.

Ok, so you have dueling and thrown weapon and are wanting to know what the most optimal way to stack bonuses are, possibly by using two weapon fighting.

So first, the general consensus is that you need to make 1 attack while wielding 2 weapons to make second weapon attack BA. And obviously you won't get dueling bonus for any attack made while wielding 2 weapons.

So using 1d6 thrown light weapons and 18 STR for math, your best option is:

Draw weapon, draw and throw second weapon, repeat, TWF BA throw first weapon (dueling): 1d6+6, 1d6+6, 1d6+4 = 3d6+14 (24.5)

Downside is it leaves your hands empty at end of turn. Drawing weapons next turn is no problem, but opportunity attacks can only be unarmed. Can mix it up by melee when you have 1 weapon and get same damage.

So I can do it, but at the cost of the Dueling Fighting Style bonus damage for all but one attack.

Might as well just only have Thrown Weapon Fighting Style, as using the increase to get a higher attribute would yield more overall damage by 1.

I think you can get Dueling for two attacks. Start with one Dagger (or draw one) and throw it (duelling). Draw a dagger with your object interaction, then draw a second dagger and throw that (no duelling, but TWF trigger), throw remaining dagger as bonus (dueling). Most of the time you will be empty handed for opportunity attacks. 

I don't like it - I think dueling should not attempt to be combined with any form of two-weapon fighting - but I think it works. I feel like just returning to the two dagger attacks per turn might be better. Both get duelling and thrown bonus, you can hold a shield or other useful object in your other hand and allows you to find some other uses for your bonus action. 

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. You don't add your ability modifier to the damage of the bonus attack, unless that modifier is negative.

If either weapon has the thrown property, you can throw the weapon, instead of making a melee attack with it.

start with two daggers or handaxes. Throw one or two with the same hand (2 attacks and thrown weapon- draw weapon as part of attack) gaining the thrown weapon +2 dmg bonus, move to the target and take the bonus TWF attack with the +2 dueling bonus.

Once you attack with the first weapon, you can then take the second attack even if you throw the first or the rule wouldn't say you can throw either weapon, only the second weapon, and once you attack with the light weapon you have triggered the bonus action attack even if you don't have the weapon in your hand anymore.

If you're really looking to maximize damage with a thrown weapon fighter then get Sharpshooter instead of Dueling and use darts. This will also let you ignore cover and disadvantage at long range. Overall making thrown weapons more viable. Also since you can draw thrown weapons as part of the attack action with the thrown style you can always draw a dagger after all your attacks so that you're not empty handed at the end of your turn. Granted, you do lose out on the bonus action attack unless you do dagger, dart, BA dagger. 

Another alternative is to Dual Weilder and fling javelins instead of daggers. You'll probably want an efficient Quiver though. You can always draw two at the end of your turn to maintain that +1 AC. (Dual Weilder allows you to draw two weapons when you'd normally only get one)

A soul knife can replicate two weapon fighting and also throw those weapons. They add their modifier to the bonus action attack made with the psychic blades feature.

the psychic blades manifest when each attack is made individually and then immediately disappear. This qualifies them to be able to be used with the dueling fighting style.

you can also make use of the thrown weapon fighting style on top of that, as the blades are treated as thrown weapons and can make ranged attacks.

Id probably multiclass into fighter for the first style though, maybe take a feat for the other if you really want it.