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:

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.

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:

In any piece of writing, context matters. If a rule has multiple sentences, they're meant to be read together. For example, the first sentence of Divine Sense is meant to be read with the rest of the feature's sentences, which explain that first sentence.

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.

You cannot use the wand more than once per turn with these two features

Using a wand is a special action defined by the wand's description. The prototype wand says:

As an action, you can cause the magic wand to produce the cantrip [...]

You cannot produce a cantrip from a wand with the Attack action.

When you take the Attack action, Arcane Armament allows you to attack two times in that one action. Attack meaning to attempt to hit something with a weapon. But you cannot replace an attack with another action. And using the wand to produce a cantrip is not considered, in and of itself, to be an attack. Cantrips may have attacks as part of their effects, but producing a cantrip is not itself an attack.

When you produce a cantrip from the wand you have to use your full action to do so, which means normally giving up the option to Attack.

So, you cannot take an attack given to you by the Attack action and use it to use the wand to produce a cantrip.