We rented an apartment on Airbnb for our families this past July - we had 3 adults and 3 kids and needed at least 3 bedrooms. Obviously we had bigger needs than yours, but it worked out well because we also wanted a kitchen and a room to hang out in that wasn't our beds. It was someone's home that they rented out, and so I think fit the rules for legal rentals in Paris. There were definitely more options for smaller places on Airbnb.
Also, if you have any questions about kids that age in Paris, let me know. Our daughters was 5 but turned 6 just after we got home. She loved riding the metro (her favorite part of the whole trip), going to Versailles (the gardens and Hall of Mirrors were the highlights), the Eiffel tower - especially watching it sparkle from the Trocadero at 10pm, the Catacombs, and the playground in the Jardin du Luxembourg (worth the entry cost!). She also enjoyed parts of the Louvre- the Sphinx and some of the other bigger sculptures.
We did a lot of museums because the older cousins (11 & 15) wanted to see everything and it was their first time - Louvre, Orsay, Cluny, Les Invalides. So we loaded books on our kindle and our daughter read a lot in the museums - but it worked perfectly because the adults/older kids got to see what they wanted and she was happy reading on a bench until we went to the next room. We might have made different choices of course if it was just us with her, like you will be.