I agree that "Character's Weapon" is an ugly-sounding naming, convention in general - and in this case, it doesn't even fit the established pattern of signifying an inheritable weapon that can only originate with one unit.
Though I'm not sure what a better specific replacement would have been in this case, since Ayra can't actually use Balmung. A variation of the Brave Sword would fit from the perspective of the original game, but just become confusing in Heroes as it wouldn't function like a Brave Sword. An "Einherjar's Balmung" referencing the Awakening regalia might be similarly clunky, although they sort of did that by giving Sol Katti its Awakening look.
Maybe some form of "broken sword", playing off the method to inherit Balmung without actually having the sword be Balmung?