I would not consider staying a week in Paris in July without having air conditioning - whether it's a hotel or an apartment. There are apartments that have a/c - but they tend to cost more than $300/night if you are insisting on staying somewhere that you don't need to spend much time in transit. You might have better luck with the pricing if you expand your search to include some of the neighborhoods that are further out in the double-digit arrondissements (e.g., 10th). Otherwise, look at getting adjoining hotel rooms in a modern business class chain hotel like Ibis, Ibis Styles or Mercure - what they lack in "Parisian Charm", they make up for with elevators, more English speaking cable TV channels and reliable air conditioning.
If budget concerns and wanting to stay as central as possible override your need for A/C, then you may also want to consider looking into some of the hostels in Paris and book a family room, a quad or adjoining double rooms. Many now feature en-suite private bathrooms and you will have the added bonus of being able to use common area spaces to spread out, watch tv, cook a simple meal, use the lounges for WiFi or just reading a book and having on-site laundry facilities enabling your family to pack much lighter as you can do a few loads of laundry during the week. Read reviews carefully on HostelWorld.com before you book anything to understand what other people have to say about location, noise levels and cleanliness. Some hostels are a very good fit for families, others not so much.