Your first one is questionable.
Your second two are justifiable, but awkward. They're probably the closest thing to an answer to the question of "what is the correct way to use a parenthetical clause about a subject while using the subject in the genitive?" but they're still awkward.
Your last does the best by rephrasing to make the issue go away.
So too would:
The dog's ears perked up.
In-between those two ways of avoiding the issue entirely would be:
The dog's ears (he was very big) perked up.
The dog's ears (and he was very big) perked up.
The dog's ears perked up. He was very big.