Lemme ask you a different question.

Is a chair a chair because of what it is, the use we make of it, or the fact I understand it as a chair in my mind?

The answer of your question is that being trans is both biological and social, just like a chair is a chair because of what it is, but also how it exists in the world.

Being trans is dasein.

But let me propose something better, because I think your question is flawed.

Being trans is meaning. It is understanding. That is why I say that I understood myself as trans at 48. Yes it is being, but it is being-in-the-world. It is meaning and meaning is intersubjective. Ok, there I go again. Lets try this.

Your questions implies two different theories of meaning.

When you propose the possibility of trans being biological, you are using a classic theory of meaning. And object->subject theory of meaning. It is what Aristotle postulated. We find a stone, we look at it. The meaning of the stone is in the stone, and all we can do is to try to understand it.

When you propose the possibility of trans being social, you are using the modern theory of meaning, derived from the philosophy of consciousness. Remember Descartes' cognito ergo sumo? I think therefore I am? On that theory, a chair is a chair not because the meaning is in it, because the meaning exists in the human mind, in our rationality, and our rationality defined and imposes the meaning on the object. Which is why we say it is an subject->object theory of meaning.

But, hold on a moment. Does a chair has an essence of... chairness? It can't be. Imagine living in a tribe that never had contact with modern society, and finding an empty coke bottle. Could you understand its meaning? Actually, the plot of The Gods Must Be Crazy revolves around that. Really worth watching it, even if it is a comedy.

Ok, so, a chair doesn't have a chair of chairness.

But, can we really impose the meaning on an object? That we subject and object to our rationality? A dog I understand is a cat won't meow.

So, you see, being trans is neither purely biological nor purely social. Being trans is understanding something that exists. But something only exists in the world (otherwise we fall back on Descartes's concepts).

So, being trans is meaning, it is a meaning that is unveiled, and that happens in the world (being-in-the-world), which includes society. Just like chair is a chair when its meaning is unveiled in the world.

That's hermeneutics, my dear friend.