Let's watch how the word not likes to jump around inside a verb group as we change it.
Here's a verb group with a modal verb and some tense helpers:
We could have been staying here.
If we change this to a negative statement, where does not go?
We could not have been staying here.
Now let's remove the modal verb could and see where not goes:
We have not been staying here.
And if we change the tense:
We are not staying here.
And if we change the tense again:
We do not stay here.
We get our do tense helper creeping in.
But some rules are made to be broken. Breaking this one is great for adding a little bit of judgy sarcasm!
We could have not been staying here...