Boutique Hotel

As in your first thread, we need to know how many rooms you need, and what the 400 covers. One room for 6 people is not likely. I have no idea why American dollars when you are traveling to Europe.

You really need to clarify the budget window, preferably in euros. Specifically, how many rooms do you need and for what general dates.

If you mean $400 per night per person, I could suggest the Relais Christine or Hotel d'Aubusson in the 6th, just across the Seine from le Louvre.

Charming little hotel in the 2nd. https://hotel-britannique.fr/

1€ = $1 (US) these days, so it does not matter which one!
What matters more is the time of the year, as hotel rates do vary.
And yes, if your budget is for three twin rooms for 400€ total, that's very tight in Paris.

My top 2 boutique hotels are :
Hotel Saint-Petersbourg Opera & Hotel Lyric Opera

Yes, dates and how many rooms you want for that 400€ per night is critical information. I mean 400€ per person versus 400€ per night for 6 are worlds apart in terms of what you could get. I would normally suggest Hotel Excelsior Latin as a good budget hotel if you are looking for two triples for that 400€ but it is currently closed for renovation. It's sister hotel, the Hotel Michelet Odeon might be in budget for two triples for roughly 400€ but it really depends on your dates. There are two other sister hotels for the Hotel Excelsior Latin. You can find them at the first link that I included but note that the Hotel Notre Dame Saint Michel does not have triple rooms.

Mayfair Hotel near Place de la Concorde. Small boutique with marvelous breakfast.