-If theists believe something that does not contradict secular
morality, then anti-theists have no justification to say it's
irrational and immoral.
-This is because they "just believe" in something based on "irrational
intuition", and if it does not contradict secular morality, then it is
permissible.

Let's apply this to the stance of the anti-theist... that religion, in
and of itself, is immoral. Let's pretend that anti-theists have
arrived at this conclusion via the same "irrational intuition" that
theists are allegedly using for their conclusions. Of course this
isn't actually true and anti-theists do a lot better with their
arguments, but pretend that it is based on nothing aside from
"irrational intuition". Now change "theists" in the above sentence to
"anti-theists".

-If anti-theists believe something that does not contradict secular
morality, then anti-anti-theists have no justification to say it's
irrational and immoral.
-This is because they "just believe" in something based on "irrational
intuition", and if it does not contradict secular morality, then it is
permissible.

Now, the question remains, does anti-theism contradict secular
morality?

Clearly anti-theism does not contradict secular morality more than
theism (and of course, I believe it contradicts secular morality a
great deal less, but that's besides the point to this discussion at
hand). Therefore, based exclusively on this, there is no justification
that someone should be "anti-anti-theist" for the reasons above. An
anti-anti-theist can not consistently criticize anti-theists for the
very thing the anti-theists are criticizing theists for.

That makes anti-anti-theists hypocritical.