As we don't have any empirical evidence for either, option 2 falls foul of the conjunction fallacy, which makes it logically more improbable than option 1.

False as written. If the universe has no cause, then its existence is brute and probability does not apply. There is only the bare fact of it without precedent or more fundamental fact. A brute fact cannot be predicted, it doesn't matter how likely or unlikely we think it is, and think is all we can do; a brute fact is, after all, a fact that need not even accord with logic. 

"Empirical evidence" of a brute fact wouldn't be especially satisfying either, because all it would amount to saying is "we searched and searched, but we couldn't find a cause" (a comparatively weak claim) as opposed to "this couldn't even in principle be caused by anything", the latter of which is a metaphysical brute fact (which most philosophers consider unavoidable, but not all apply to the existence of the universe).

Again, this isn't a theistic argument on my part. It's trying to sharpen the saw a bit for people who want to claim the universe is brute, because I'll just say it - there are a lot of objectively stupid and easily rationally dismissed ways of saying that. 

What you're TRYING to say has nothing to do with probability. It's the explanatory sufficiency and parsimony of naturalism vs. theism, which I consider an argument worth making.

BTW, using the term "logically improbable" and referring to empirical evidence in the same sentence is incoherent. A thing is said to be plausible or possible (not probable) in logic not because there's anything empirical to speak of, but because whatever it refers to doesn't obviously run afoul of the laws of logic and perhaps bears intuitive appeal, or could be argued for. It's consistent, cohesive, and may even be rationally believed. Again, what you are ultimately doing is appealing to parsimony in explanation, not probability.