I love this stuff.
I just used Booking.com to look for apartments.
Below is the path I recommend to get you started.
Opening screen:
Paris City Center.
Your arrival and departure dates.
4 adults.
Ignore the number of rooms.
Results page filters:
Apartments.
1+ bathrooms.
5th and 6th arrondissement.
8+ Review Score.
After that you can do more filtering and sorting. Using stars (*) may make your results smaller because the star system is not typically used for apartments.
I tested this using 8 May - 22 May for the dates and ended up with 26 listings. That isn't very many. That's probably due to the large number of people and the fact that May is not that far in the future as these things go. Your real dates are likely to produce a different number of results.
After that, it's a matter of looking at each apartment one at a time to see if it will work for your group. With a potential of so few options, I would not filter any further.
The test results included both what I think you meant by French, French, French and more modern styles. Personally, I consider anything in France to be French.
When you look at your results you will probably see many listings that have older architecture with modern renovations.
I was surprised at how many listings were in apart'hotels that have some features the other listings don't. I was also surprised to see that some places had airport shuttle service, usually for a fee. That could be very useful for your group.
Although all slept 4 or more, not all were exactly what you described. The prices for 14 nights ranged from € 2,800 (+€224 for taxes and charges) to € 19,436 (including taxes and charges). About half were below € 4,000 (before taxes and charges) and about half were above.
Price didn't always coordinate with the number of baths or beds, but the 2-3 bedroom and 2 bath places were in the € 4,000 - € 9,000 range. There was even one with 4 bedrooms. The higher the price, the more likely the taxes and charges were included.
Having stayed in apartments all over Europe where the cost for me alone was about € 1,000 per week, the prices I saw seem pretty reasonable to me, especially when you want to have all of you in the same apartment.
Some features cannot be accurately found by filtering. You have to look at the complete list of facilities, read the reviews and look closely at the pictures. Is there a clothes washer? Is the fridge an under counter one or a tall one with a freezer. Is there a microwave? An oven? A dishwasher? How do I get rid of the garbage? Recycling? What's the square footage?
My personal deal breakers:
Having to clean the place before I leave. (I'll happily pay for the cleaning.)
A tub without a shower head on the wall, hand held or stationary.
Rain showers with no way to get away from them in the shower space.
Have fun searching and browsing. And have a great time on your trip.