Paris Budget accommodations in the Rue Clar , 7th Arrondissement

In case you don't know this: Within the last few years Paris has been cracking down on illegal (unregistered) apartment rentals. If you're looking for an apartment rental instead of just a bedroom in someone's home, be sure the property listing includes the 13-digit registration number. As enforcement actions continue, more landlords will be assessed very high fines and will immediately pull their properties off the overnight-rental market. You don't want that to happen to you shortly before you're due to fly to Paris! Note: I don't know anything about HomeAway; perhaps it adheres to all the requirements and lists nothing but legal rentals.

Check this place out: https://www.facebook.com/pg/Locationparis9-260762077394544/posts/?ref=page_internal also http://locationparis9.canalblog.com/
it is a lovely suite in the bottom floor of a mansion near Montmartre. Bus across the street, Metro a couple blocks. Near Rue des Martyrs. Afternoon market on Friday. The pricing is amazing. We stayed in June for E420 for 8 nights. The hostess Nadine is wonderful and helpful. It is near Montmartre, but out of the craziness on a street full of apartments and businesses. I have stayed here couple of times and really think it is good value. Also it is close to Gare du Nord, so easy access from the airport.

Check out Hotel Beaugency, we have stayed there twice.

Note that September has fashion week and conventions and therefore can be highly booked/ expensive.

The Loire to the South of France tour is fab-u-lous and best wishes that for lucky RST #7 your guide will be Virginie. Mid May 2018 I chose charming (if snug) and nicely updated Hôtel Mistral (petit déjeuner-inclusive) in the Montparnasse (14th) arrondissement to make catching the train to Chartres that much simpler/easier, for our extra Paris (2nd visit) time prior to tour start. Paid €123/night for superior twin room. Rue Daguerre is a nearby market street, in a similar vein as Rue Cler in the 7th. We knew we wanted to tour the Paris Catacombs and being close to the Montparnasse Cemetery was another plus. Taking in a flea market, Marche aux Puces de la Porte de Vanves was a highlight. Having Métro stop Gâité close to the accommodation made getting around to other Parisian arrondissements a breeze. We chose to taxi from CDG (set €55 Left Bank fare) on arrival but knew we could have opted for Le Bus Direct (€17 pp) with its Rue du Commandant Mouchotte at Gare Montparnasse drop off and walked the 6 to 8 minutes it would have taken to reach our hotel. Note, while I knew to get in the official taxi queue at the airport, we were quickly approached by a rogue driver who smoothly attempted to direct us away from the rank. It took me a few steps to recognize I was about to do what I shouldn't and said, "Non, merci" and went back the proper line. Enjoy your travels!!

Just know that Rue Cler is a bit out of the way from the central sights of Paris. It's fine but will not be super central.
If you look at booking.com, you can set the price on the left hand side of the screen and then search on a map for things within your price range. There are plenty of places in Paris to choose from.
Just know that September is a high-demand time for hotel rooms in the city. Lots of business conferences and, I believe, fashion week falls in there somewhere.
Good luck!