Place d'Italie would be a much better locale for a family IMHO. I doubt that you'll find much of an apartment for $80 a night through any of the services-I'd head to VRBO or a similar site, and really so your homework since just about anyone can put a place up there.

Also, since you're on a budget, I'd get into the city via the Roissybus with a transfer to the Metro. It'll be a little tough with luggage, but spending money on a cab or even the AF bus is money better spent elsewhere.