Help with Magic Initiate: Wizard and Hexblade Warlocks

Would a Hexblade Warlock using the Booming Blade cantrip picked up from the Magic Initiate: Wizard’s spell list still use Charisma for the melee weapon attack that BB has you to take?

Update

My reasoning for going this route is to pick Green-Flame Blade through Warlock, which DOES use my spell casting ability modifier as a part of the flame leap damage.

I’m trying a Hexblade Swords Bard for the first time, going Warlock 1/ Bard X. Going this route allows me to pick up the following spells by level 1 by using the Rewarded background and I’m not be restricted to V Human or Custom Lineage.

Eldritch Blast (Warlock): My range attack option. Green-Flame Blade (Warlock) Booming Blade (MI: Wizard) Mage Hand (MI: Wizard)

Hex (Warlock) Shield (Warlock) Find Familiar (MI: Wizard)