No, this won't work exactly the way you describe.
Two-weapon fighting requires two weapons to activate
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.
The rules for Two-Weapon Fighting require you to be holding both weapons at the time you use it. It say you use a weapon that you are holding in the other hand. Not you will be holding. Two-Weapon Fighting requires you to be holding two weapons to use it. It isn't called One-weapon-and-maybe-another-later Fighting for a reason ;).
That means you cannot use Two-Weapon Fighting unless you are holding both daggers at the time of the attack.
This question talks more about this specific issue.
If you start with both daggers in hand it works
If you start with two daggers in hand the order would go like this:
(Which hand you use right/left, doesn't actually matter. 5e does not have the concept of an "offhand". So, you can flip right for left below easily.)
- Attack action to do one dagger attack (right hand)
- Drop dagger (right hand)
- Grapple using Extra Attack (right hand)
- Use bonus action to perform TWF attack with other dagger (left hand)
There is no reason you cannot take your bonus action TWF attack after your Extra Attack grapple because the rules say:
You choose when to take a bonus action during your turn, unless the bonus action’s timing is specified...
In fact, this is the default place for the TWF to go. So there's not much to debate here. Regardless, since the TWF attack is a bonus action and the timing is not specified, you can take it any time during your turn.1
Also, since you started with two light melee weapons in your hands you meet the requirements for TWF. It doesn't matter that you drop one subsequently.
1 - It is unclear if you can technically insert the bonus action TWF attack between the 1st and 2nd attacks though according to Jeremy Crawford here. This isn't relevant to this case though. (thanks @V2Blast for pointing this out)