What is the interaction between Empowered Spell and Booming Blade or Green-Flame Blade?

Question

Can you reroll the damage dice that come as part of the melee weapon attack? That is, is the melee weapon attack part of the damage for the spell?

Empowered Spell reads:

When you roll damage for a spell, you can spend 1 sorcery point to reroll a number of the damage dice up to your Charisma modifier (minimum of one). You must use the new rolls.

Booming Blade and Green-Flame Blade read:

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, otherwise the spell fails.

Answer 1

Yes, the Melee Attack Damage counts as part of the overall Spell's Damage, and therefore can be affected by Empowered Spell

The first thing we can say about this interaction is that, intuitively, the weapon attack is something that is done as part of the spell. If we assumed, for the sake of argument, that booming blade was being cast by a level 3 Sorcerer against a target that never willfully moves (thus never triggering the secondary effect of Booming blade for the 1d8 Thunder Damage), it would be pretty strange to argue that the spell simply "does no damage", on the basis of the only damage dealt having come from the melee attack itself.

So in this context, it seems like the melee damage counts for "rolling damage for a spell".

Secondly, there's textual evidence to support the idea that this melee damage counts as part of the overall spell damage from the section about how the spell gets stronger at higher character levels:

This spell's damage increases when you reach higher levels. At 5th level, the melee attack deals an extra 1d8 thunder damage to the target, and the damage the target takes for moving increases to 2d8. Both damage rolls increase by 1d8 at 11th level and 17th level.

―Booming Blade, Sword Coast Adventurer's Guide, pg. 142

The wording "this spell's damage increases" implies that the original melee damage is indeed being included as part of the overall spell damage.