Dear "Hindu Atheist". Saying "my religion gives me permission to not believe in god" is like saying "my jailer gives me permission to not remain in prison". Atheism is a resistance, not only against the phenomenon of religion, but also against religious authority.
You are trying to have your cake and eat it too. You are trying to claim to be modern and rational while continuing to hold on to your Hindu privilege. Using the Atheist label to uphold a millennia old system of religious authority is an idea that is completely in sync with all else that the modern savarna does - hijacking the language of resistance to try and use it in the service of oppressive systems.
That's where "Hinduphobia" came from. That's where all the wailing about "Hindu khatre mein" came from. These are all instances of reframing the slogans of the oppressed to serve oppressive systems.
As far as Atheism is concerned, nobody needs permission from a religion to be an atheist. Modern Atheism arose as a resistance to religious hegemony and authority. To say it is "allowed" by your religion is to strip it of its essence. That kind of Atheism is just another Hindu privilege.
If you only use the phrase Hindu Atheist to differentiate yourself from contemporary European Atheism (focused on personal gods and exclusivist monotheism), you can always call yourself an Indian Atheist or a plain old rationalist.