The rules for two-weapon fighting state:
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.
Emphasis mine. Now, under most circumstances you would wield two light melee weapons, attack with at least one of them during your Attack action, and attack again as a bonus action. However, the rules themselves just say "take the Attack action and attack with a light melee weapon", not specifically "take the Attack action to attack with a light melee weapon" or "attack with a light melee weapon during an Attack action".
In other words, both the requirements "took the Attack action" and "made an attack with a light melee weapon held in one hand" need to be met during a turn, but the light melee weapon could be used to attack separately from the Attack action. Is that correct?
An example for how to accomplish this: An Eldritch Knight Fighter takes the booming blade cantrip and the Dual Wielder feat. That Fighter takes the Attack action, attacking with a Greatsword, then uses Action Surge to take an additional action. He drops the Greatsword (everything I've seen indicates that dropping a weapon doesn't even require an object interaction), then draws two light melee weapons as permitted by Dual Wielder. For the second action, the Fighter casts booming blade (SCAG, p. 142) which states:
As part of the action used to cast this spell, you must make a melee attack with a weapon against one creature within the spell’s range
Thus, after those two actions the Fighter would have both taken the Attack action and made an attack with a light melee weapon in the same turn, and I would expect he could use two-weapon fighting to make a bonus action attack with the weapon not used during booming blade.
Note that this scenario is just an example; I'm not 100% certain there's a way to cast booming blade while holding two melee weapons, but I know there are other ways to make an attack outside of the Attack action and I just couldn't find one quickly that was more definitively possible.
Best Answer
The attack must be made as part of the Attack action
I can see your reasoning, but at the very least that goes against what I believe the intent of the wording is. The obvious reading of the rule is that you attack with the Attack action using a light melee weapon, then you can use your bonus action to make another attack with your other light melee weapon.
The rule you're referencing:
I read this to mean that they must both be done at the same time. You have mentioned other ways this could have been worded to achieve this intent that would not provide you with your loophole (Action Surge and Booming Blade, etc) but this is the wording they went with, and I believe their intent was for "and" to mean "both of these things must be done as one action", although it doesn't explicitly say that.
Although not exactly about this, Jeremy Crawford has said the following, which broadly covers my reasoning presented here:
Hence when you take the Cast a Spell action instead of the Attack action to make the melee attack with a light weapon, that does not satisfy the "and" part of that clause.
Without clarification on this exact issue from a designer, this is probably the best I'm going to be able to conclude.