VRBO: Family of 5. Paris neighborhood. Easy train commute to City

Well not sure how you think you are living like a local.. but the reality is September is busy and many good places are booked up .
Do not stay outside the city.. the time and money wasted are not worth it for such a short visit.. staying outside Paris is a minimum of 30 minute commute each way, and tickets are more expensive( you can get any of the commuter deals either ) .. so better to spend a bit more and stay inside city limits.

With teenagers just get two rooms.. you can likely find perfectly acceptable places to stay for 100-150 euros a night each room.

Keep in mind many apartment rentals are in fact illegal also..I wont go into details, but people have been left in the lurch due to last minute cancellations ( when the city nabs an apartment owner for illegally renting ) .. There are legal rentals , however you have left it a bit late to do the research . Even apartments rented by reputable type agencies can be illegal.

You can book rooms that sleep 5 people in Citadines or Adagio apart'hotels.
You will have the same kitchen as a rental apartment will have, laundry services on premise, and maid service once a week.
I would recommend staying in the 11th arrondissement, at Citadines Richard Lenoir. It's a convenient neighbourhood for walking or transportation options, and the outdoor food market at Bastille will let your kids pretend to feel like locals. What Citadines may lack in charm will be erased as soon as you step out into the neighbourhood.
You can get anywhere from Bastille in 15 minutes by Metro, but I would not stay outside of the periphery of Paris.
There are also Citadines and Adagio at different locations at different price points in Paris.
As pat said, you won't be able to get to the major monuments in 15 minutes. Even if you took the first train - around 6 AM - It would take about 30 minutes to get to the centre of Paris. It would also make for a very long day, if you get tired and need a nap.
Beware that vacation rentals prefer that people stay for 5 or more days, and the owner or manager may cancel your reservation last-minute in favor of someone who will stay longer.

Try renting an apartment from the company Vacation in Paris. They may have something available for you. We're renting in late September with them. They're a US based company, and may have some vacancies left.