Room rates in many cities--Paris certainly included--are highly seasonal. Can you be more specific than "fall"? It has been decades since I spent the night in downtown Paris, but I know folks have said here that September is the busiest, most expensive month of the year there.
How old are the children? That might matter, too. Booking.com requires the ages of children before showing any results.
I use booking.com a lot; have you tried it? There are lots of useful filters. One thing to watch out for in your price range will be apartments with just one real bed for two people, with the other two expected to sleep on a sofa bed. Those things are sometimes really uncomfortable.
I suppose the Ibis chain is reasonably equivalent to Premier Inn. "Ibis Budget" hotels are less expensive than the plain "Ibis" hotels. "Ibis Styles" are somewhat more expensive. I don't know that any of them have rooms that sleep four people.