How to remove window screen to clean exterior facing glass?

The screen has pull release tabs at the top, but where it would have to lean backwards to come out, it’s blocked by the bottom of the bottom pane (which has to be slid up to access the tabs). I am newly renting here and the landlord says they won’t send someone to remove the screen for cleaning since it isn’t broken, even if it is hard to see out of the window because of how dirty it is.