That's easy.
You give them four arms.
No special rules or mechanics needed.
I have four arms, ergo I can cast a spell with one, use a shield with another, stab you with my shortsword with the third an use my bonus action to stab you with a dagger with the fourth.
Whether or not you choose to allow it, the rules have to address it.
... I don't see how that's a problem?
You can cast a spell with the Cast a Spell action, you could attack with the Attack action, you could even Action Surge to do both and have it play out exactly as you describe. All without any mechanical changes.
I'd say some kind of Vryloka/Dhampir style race would make the most sense.
The rules as written are for two-armed characters.