God exists outside the laws of science that we exist in. Heck, he created them.
That means that in the universe God created for us, a rule is that something cannot come from nothing.
You're begging the question but let's set that aside.
You assert that the laws of science apply to the universe, which I think we all accept. But you're using this argument to reach beyond the universe, that does not apply.
Big bang cosmology begins with a single point where everything in known existence densely resided. This is called the singularity. In science, a singularity is a point where the laws of not just of physics but also math and logic break down. What big bang cosmology asserts is that it was during the big bang explosion where the laws of the universe themselves came into existence. So in other words, much like your assertion of God, the singularity is not subject to the laws of the universe.
Therefore there is not one single problem you have raised for which God answers and cannot be applied to the singularity. You say God always existed, but you disregard that possibility towards the singularity. That's special pleading.
Here, let's take a step back from big bang cosmology and look at it from a much simpler perspective; why is there something rather than nothing?
You assert God. Ok, so is God a something or nothing?
If God is nothing, then he cannot logically be asserted as the cause for everything as that is self contradictory.
So God must be a something. Your answer to the question is this that the reason there is a something is because of something. But something cannot cause something's own existence, so "something" cannot be the answer to this question.
There is no third option here, as nothing is simply the negation of something.
So in other words, just like the singularity, there is no logically valid answer to this question. The proper response is therefore "I don't know".