Okay, so I understand your position better now, I think. What I don't understand is how it follows that this position precludes the Cosmological argument. Perhaps if I rephrased it as such:
If the universe began to exist, then it has a cause
The universe began to exist
Therefore it has a cause.
The evidence for "if the universe began to exist, then it has a cause" is the inductive argument "everything that I have seen that begins to exist has a cause." The evidence for "the universe began to exist" is the Big Bang Theory. The Big Bang Theory in this case is merely evidence for the evidence, and is in no way in any sort of "direct competition" with "God exists". I don't think it violates your principle.
Evidence 1) Anything that begins to exist has a cause
-Support: everything that I have seen that begins to exist has a cause.
Evidence 2) The universe began to exist
-Support: The Big Bang Theory (every time I type this, I keep thinking about the TV show. Okay, sorry, just wanted to say that.)