OK.The person making the existence claim (a claim that something exists outside of what is normally encountered in life) has the burden of proof.
OK.To be rational is to ratio your belief relative to the strength of evidence. Extraordinary claims require extraordinary evidence.
Oh, I rather think there is, it's just that the 'evidence', that is the fruit of a rational discourse, is itself extraordinary..There is no evidence of god, therefore it's irrational to believe in it.
God is, by no measure, 'ordinary' nor the object of what most would describe as 'normal' and 'everyday', or mundane, experience.
So I rather think the flaw here is your supposition that for God to exist requires the ability for you to prove to your own satisfaction that God is something which, by my understanding (and mine is common to almost all theologies), God is not.