Booking.com is finding 68 properties in Paris with an air-conditioned room costing no more than 200 euros per night for your time period, but the locations are not necessarily central. I don't know Paris, but I recognize a few of the areas as probably farther out than you'd want to be. When I add the "Paris City Center" filter, the number of hotels listed drops to 2, one of which is offering two twin-bedded rooms rather than one large room. The second hotel has a room with four twin beds available.
Going back to the full list covering all of Paris, some of the rooms have one regular bed and one sofa bed (I would not want that for 5 nights) or one regular bed and one bunk bed (might be fine).
Increasing the budget to 240 euros per night gets you 9 options in Paris City Center (as defined by booking.com).
All rates quoted above exclude some "taxes and charges". They vary by hotel from about 20 euros to about 60 euros for the entire stay, so including them will drive up the total cost by 4 to 12 euros per night.