How do I allow access to my custom module to "anonymous user"
I would suggest you gain more control over the access, by switching to "access callback" => "my_custom_function".

Then in my_custom_function() you can write just "return true" which will make it always accessible. 

You could also add a log before "return true" so you can  tell whether the function is being called. If it isn't, then you know access is being blocked further up, before you get to that stage.