Paris Hotel Recommendation for Family of 4 (June)

What is your budget per night in euros, ideally but USD would also be great? I am not sure what amount constitutes going overboard or spending a little more for you. But taking a shot in the dark with no budget guidance: Hotel Excelsior Latin on Rue Cujas (http://www.excelsior-paris-hotel.com/en/).

Probably many more options for BNB rentals than hotel rooms for your family of 4 to stay.
Only a small number of hotels will have such rooms where any 2 bedroom Airbnb or even 1 bedroom with a living room sofa bed could work.

Hotel Latin Excelsior has 4 bedded rooms as well as a few suites for families with bedrooms.

"Only a small number of hotels will have such rooms"
Actually, not really. The OP prefers two rooms OR a family room. I think one would be hard pressed to find a hotel in Paris that could not offer two rooms for a family of four.

I'd suggest either Citadines or Hotel Residence Henri IV.
Citadines is in close proximity to the Bastille market and has good transportation - in the 4th arrondissement.
Hotel Residence Henri IV is in the 6th arrondissement and has lots of things to do and good transportation.
Both have kitchens.