Both are grammatically correct, but in conversation or in practice we'll use the second sentence because "being late" is equal to "my being late" or it is understandable that about whom we are talking.

They are both natural and understandable sentences, and I would expect either of them to come out of the mouth of a native English speaker.

However, when you use a possessive for an action like this, it adds an implication of moral blame or responsibility.

This becomes more apparent if you put in a context where the person isn't necessarily angry. So as between

"we missed the bus because you were late"

and

"we missed the bus because of your being late"

the second one has a stronger note of blame or accusation, that the person was responsible for being late -- it's making them "own" that action. The first one, depending on the way it spoken, might be more neutral, that the person just was late because of circumstances.

This is true even when it's reflexive, so "we missed the bus because of my being late" sounds a bit self-admonishing, a little "yep, I messed up" to it.

Those moral-ownership possessives also sound a bit more informal, and less professional.