I personally have a problem with an atheism with moral absolutes. I can't see how that works. But even if I didn't have that problem, I'd have a problem with the taking as given that duty and happiness are seen as opposed in Christianity.
Somewhere in this thread Rand rightly contrasts doing something on a whim (because the mere doing would make one happy) and doing something with the object of happiness. THAT would be our point of contact, or one of them.
It simply does not, it is a formal fallacy of logic. Any atheist who says I am immoral is no different than a preacher or rabbi saying I am a sinner.
As far as philosophy goes, Moses was the first who said our rights are not dependent upon the whims of an earthly monarch.
Like Jefferson in the Declaration of Independence ...
"...to which the Laws of Nature and of Nature's God entitle them... that all men are created... Assembled, appealing to the Supreme Judge of the world... with a firm reliance on the protection of Divine Providence..."