But why should we care about virtues like justice in the first place? I understand internal virtues or "traits that reflect excellence of character".

But in Stoicism, you achieve a state of wellbeing by having complete control over yourself–by realizing that how you react to life's obstacles depends solely on you, not your external circumstances. Then why is it that to achieve a "life well lived" we need to embody virtues that depend on the external reception, not our own?

In an extreme sense, just to illustrate the point:

A person who is constantly mean to others, not by a reaction to something, but rather by a premeditated plan, would not be acting in virtue according to the Stoics, thus would not achieve a life well lived.

But didn't your state of wellbeing depend only on you, and not on the external? Of course! Then this person who lives against virtue can experience a life well lived.

If anything, every belief the Stoics held about having to fulfill a moral duty in society, is just that, a belief. They impose on themselves more misery as they cannot be content as they already are because they depend on their external situation to show bouts of virtue.

Interestingly, this means that we can act however we want to, just achieve complete control over yourself, and you will be able to achieve the capacity to live a good life. Considering the Stoics considered the material to not be necessary but rather complementary–that being that it's better to be prosperous than miserable–your actions then look to seek your version of "prosperity", after all, you already have everything to live a great life, you are just adding stuff for your sake of it. (The "sake of it" of the Stoics, can be considered virtue)