How does the Trickery Domain cleric's Invoke Duplicity feature interact with the Booming Blade spell?

Question

How does the Trickery Domain cleric's Channel Divinity option Invoke Duplicity (PHB, p. 63) interact with the Booming Blade spell (TCoE, p. 106)?

Does it allow Booming Blade to be cast from a distance?

Answer 1

The effects of booming blade occur as though you were in the illusion’s space.

Invoke Duplicity states:

For the duration, you can cast spells as though you were in the illusion's space

Naturally, this means the effects of the spell occur as though you were in the illusion’s space. The effect of booming blade includes:

You brandish the weapon used in the spell’s casting and make a melee attack with it.

We cannot treat the melee attack of booming blade as though it were separate from the spell's effect, since the rules for spellcasting state:

The rest of a spell entry describes the spell's effect.

The melee attack described is part of "the rest of a spell entry", so it is part of the spell's effect. Ergo, "you brandish the weapon used in the spell’s casting and make a melee attack with it" as though you were in the illusion’s space.

This is the same logic applied in the accepted answer to this question, Can Invoke Duplicity use touch spells?. If Invoke Duplicity does not work with booming blade, then it does not work with any spell that requires you to make a melee attack.

Answer 2

If both you and the illusion are within 5 feet of the target, you get advantage on the attack

Invoke Duplicity says:

For the duration, you can cast spells as though you were in the illusion's space, but you must use your own senses. Additionally, when both you and your illusion are within 5 feet of a creature that can see the illusion, you have advantage on attack rolls against that creature, given how distracting the illusion is to the target.

Booming Blade says:

You brandish the weapon used in the spell’s casting and make a melee attack with it against one creature within 5 feet of you. On a hit, the target suffers the weapon attack’s normal effects and then becomes sheathed in booming energy until the start of your next turn.

There is no other interaction effect: because you must brandish the weapon and attack with it, it does you no good to cast the spell as if you were in the illusion's space. You still must make an attack with the weapon, and the weapon is not in the illusion's space.

Invoke Duplicity only allows you to cast the spell as if you were in the illusions space. While targeting could including Self be considered as part of casting the spell, making an attack with your weapon clearly is not part of casting any more. It is part of the effect of the spell, once it is cast, as a separate step.

Answer 3

Booming blade does not interact with Invoke Duplicity

The illusory duplicate cannot make the attack that the casting of Booming Blade requires because:

You brandish the weapon used in the spell’s casting and make a melee attack with it

If the target is not within your reach, you cannot make the required melee attack.

Answer 4

I may have figured a way for you to do it. I would first make the argument that a magic sword is worth at least 1 silver piece. The description of Shadow Blade states that the sword used is a "magic sword" with physical properties unlike the Flame Blade sword. You could argue that you can cast Shadow Blade from your illusory duplicates spot and attack with booming blade or green-flame blade that way.

Alternatively you would need to multiclass into 14 levels of illusion wizard for the illusory reality feat. You could create an illusion of a dagger or sword in your duplicates hand using something like silent image and then make it real to use as part of the attack even though the weapon itself can't deal damage or harm someone, the attack can still land discharging the spell onto it's intended target.