Reasonable budget for an apartment for 5 nights in Paris for a family of 4?

Reasonable means different things to different people, and budgets vary widely.What amenities, services, quality are you looking for?

Nothing super fancy....safe neighborhood, a kitchen, beds to accommodate 4 people comfortably, and a/c.

You might want to expand your search beyond the river and Latin Quarter. Paris is a walkable city and has a decent Metro. If you can expand your search it might show better pricing.

I agree I would expand my search if you want quality beds (vs pullout) and at a lower price.

This one by Vacation in Paris may fit your needs. Great company, have used them several times.

https://www.vacationinparis.com/listing/215/luxembourg---st-germain-two-bedroom-delight

I suggest you look at VacationInParis for your rental. This is a reliable, long-established company, based in the US so communication is easy, and prices are quoted in $$.

This one-bedroom flat in the 5th (Latin Quarter) in available in late June and is $250 a night, so under $2000 for a full week. Less if you stay the 5 nights you mention. The sofa bed in the living room can be made into two twins.

https://www.vacationinparis.com/listing/382/latin-quarter-dream-one-bedroom#checkavail

If you prefer two bedrooms, this flat in the Marais is also $250 a night:

https://www.vacationinparis.com/listing/172/marais-paris-glamour-two-bedroom

Or start your search on the main page for more results:

https://www.vacationinparis.com/index

Are you looking for air conditioning? Summers is Europe have been dreadfully hot these past few years.
Safe travels!

I suggest you look at VacationInParis for your rental.

Does anyone see any registration numbers on any of the Vacation in Paris listings?

It appears that this company blocks their website from access within France. They’re not trying to hide from the rental apartment inspectors, are they?

I have talked with Vacations in Paris in the last year about their rentals. At the time, some had registration numbers and some did not. The staff member was up front with me about it when I asked. She then sent me a list of their properties that had the registration number. I did not end up renting from them but I have rented from them one previous time and was satisfied with them.

I confirm that I cannot access the vacationinparis website from Paris where I am right now. Hmmm...

Yes, that seems pricey. We stayed in this place, 4 of us for 4 nights, I think it was $500 a night but we were in their largest or second largest apartment. There are other options that will sleep 4 in beds (when 2 of the 4 are kids).
https://www.leprinceregent.com/
Also, look on booking.com, filter by apartments with ratings above 9 in the city center. You might keep the dates open just while you're looking because it's possible some haven't opened next June yet for reservations, but it'll give you a good idea on pricing and what you get.

The staff member was up front with me about it when I asked. She then sent me a list of their properties that had the registration number.

What is required is quite clear, short term apartment rentals must have the registration number clearly visible on any advertisement. Not only is this an indication that the rental is potentially not legal, reassuring for the client, but city inspectors check these as a quick indicator of fraudulent rentals.

I might approach Vacation in Paris with caution.

If having a registration number is important to a vacationer, approach every agency, not just Vacation in Paris, with caution. I have not seen an agency renting apartments in Paris that has a registration number for every apartment listed.

This last line of the original post raises some question for me. Why do you want an apartment? Do you think that you will get more space or do you want the option of preparing meals or snacks? Have you been to Paris before? Do you think that you might need a front desk to give you some assistance during the trip? Do you realize that most apartments will not have AC and will likely have only one bathroom? I can't really say if the apartment that you mention is overpriced because we don't know the finishes, the exact location, the number of square feet, or whether it has AC, etc. But, it's not a binary choice between the $600 per night apartment and the very mediocre Hôtel de la Eiffel. There is a world of other choices out there. It's also very early to be booking now for a June trip.
The Latin Quarter near the river is tourist central in Paris so expect high prices especially during one of the most expensive months of the year. I just did a search on AirBnB using June 4 to June 10 and four persons and there was one apartment in the Latin Quarter available and it would cost US$3,853 before taxes. It has a registration number and two bedrooms and 1.5 bathrooms. Given that it appears to be being rented by an individual, I would guess that a place being rented by an agency would be more in order to account for the agency's percentage. You should be cautious when renting apartments in Paris whether from an agency or a listing site.

I have not seen an agency renting apartments in Paris that has a registration number for every apartment listed.

Renting agencies do not just rent apartments to short term vacationers. Some ground floor apartments do not require registration. These agencies also handle 1-year or longer leases and mobility leases, which are leases of as short as 1 month but are not available to tourists.

Logically, not every apartment in the inventory needs a registration number, just those that would likely interest posters on a travel forum.

Look at Apartments du Louvre. We're staying at one of their 2 bedrooms units for 5 nights for 2200 euros the beginning of June next year. That was about as cheap as I could find with a/c, 2 bedrooms, and that seemed like an approved/legal listing. I also required a good cancellation policy which was very hard to find.

I second Apartments du Louvre, https://www.apartmentsdulouvre.com, stayed there Nov/Dec 2021. I am also a Cobblestone Paris user.

Hope all works out.