This is very related to this question: Artificer Armorer Guardian attack and cantrips like Booming blade
This is to talk about the new rules introduced in TCoE with respect to Booming Blade and Green Flame Blade now needing "a melee weapon worth at least 1 sp" as the material cost (whereas before TCoE I believe it only needed a melee weapon)
Specifically, when taking the Armorer subclass for an Artificer can you use the cantrips Booming Blade and Green Flame Blade with the Thunder Gauntlets provided by the Guardian Armor Model?
This is intended to talk about whether or not properties of the "base armor" used for the Arcane Armor be transferred over to the extended parts of it (gauntlets, boots, helmet). There is a thread that talks about the magical properties of the "base armor" and whether or not that is transferred to other parts as well.
So the way I see it there are 2 interpretations:
Properties of the base armor transfer over - Therefore your Thunder Gauntlets will have an associated cost related to your base armor and be able to use the cantrips Booming Blade and Green Flame Blade. However when using magical armor for your Arcane Armor you will not be able to infuse the "extremities" because the magical property was also transferred over.
Properties of the base armor are not transferred over - From a game mechanics perspective Thunder Gauntlets are considered separate items from the Arcane Armor and since no such item exists in the PHB the cost is undefined therefore unsuitable for the cantrips Booming Blade and Green Flame Blade. This does mean that while using magical armor as the Arcane armor the magical properties are not transferred over and the extremities are considered mundane objects with respect to infusions.
All in all, which interpretation is correct or is there another interpretation?