The important thing to recognize is that Rand didn't address this question. Maybe she would have a good response to you, maybe she wouldn't. I'm not aware of anyone besides Tara Smith trying to address it. I'm sure some people have tried to from Rand's perspective, I'm just pointing out that it's not front and center.
So the way SL answers your question would probably be different from me (even though I'm sure we would agree with what the moral implications are when we do choose to live). So don't think of this answer as replacing his, and I hope he doesn't see my response as trying to drown his out.
You're right that her ethics have whim at their basis (choosing to live can be made for any reason at all without undermining her ethics), although I imagine someone here would disagree with me. I don't think this is problematic because this doesn't undermine or ignore that there is a such thing as a fact. In that way, reason is relevant, even if there is no ethical purpose for reasoning prior to the choice to live. After all, Rand considers epistemology hierarchically prior to ethics. You would have some sensation about the world around you from the moment you were born, or sensation within you, which is at least some dim awareness that there is such thing as reality.
On some level, I don't know if Rand would accept such implications. She starts a sound a lot like Nietzsche. It says if the choice to live is equivalent to will to power. Will to power doesn't mean willing to show power over others. It means a will to show some kind of power over your individual life. There is no rational basis to such a will. If anything, we could call it will to reason, to point out that Rand still sees reasoned thinking as always relevant (distinct from rational thinking that assumes an end goal has been selected). Rand herself thought of her philosophy as a philosophy of reason first, not a philosophy of rational selfishness first.
By the way, EC was referring to rationalism (lowercase 'r') with the definition that Peikoff uses in his lecture "Understanding Objectivism". It usually refers to talking about abstractions with absolutely no effort to concretize them or ground them.
This sounds very interesting. I argued for something like this once before, but for a discussion on free will. Namely, that whatever particular method one uses to make a decision, that method will produce the same outcome if the identical context is repeated. I take that view as something consistent with Rand, but it's hard to say. In some sense, moral action is universalizable even for Rand. Of course, she treats universals as something different than Kant. Even more, Rand cares about an interested perspective and including it as a necessary part of rational action. My knowledge about Kant is limited, but I do know that he pushes for objectivity in the disinterested sense.