June trip to London then Paris...

I responded on the England Forum.

The Darlington Hyde Park has triple rooms within your budget in mid- June. It is near Paddington Station, so you could take the Heathrow Express from the airport. Get the tickets now for a better price ( your daughter is free).

https://www.darlingtonhotel.com/accommodation/

And look into tickets for the Harry Potter studio tour now if you want to do that. They sell out quickly.

Hotel Suggestions in Paris (Not sure if all will be below US$250 per night in June (budget is low for a triple in June in Paris) for a triple or will have availability)
Hotel Excelsior Latin in the 5th
Hotel Le 123 Elysees in the 8th
Familia Hotel in the 5th (I am not a fan but a lot of people like it -- location is great but I like better finishes and bigger rooms for three people)
Hotel Lorette in the 9th (I know it's not in your preferred districts but it's only about a kilometer from the eastern edge of the 8th and it has a great apartment for 3 -- of course, the apartment may not be available but if it is, it would be great)
Paris France Hotel in the 3rd (again not in your preferred hood but this place is a keeper for the price -- 4-minute walk to Republique metro stop and access to 5 metro lines)

Thank you so much--none of the hotels were available during our time line.

Buy your Eurostar tickets London-Paris from that company directly, and as soon as possible. Prices rise as departure nears.

https://www.eurostar.com/rw-en/train

At this point it is getting late to get a choice hotel at that budget in those areas.. for three.. ( and yes you have to book a triple not a double).. but I would try looking for hotels using the booking.com website.. you dont have to book through them.. just use their search engine to get ideas.. then contact hotels via their own websites to see their rates.. just take best rate( I have booked using both booking.com and hotel websites for years.. )

Your desire for a concierge likely isn't going to happen in that budget range.. but by reading reviews you can search for those that mention nice front desk clerk .
I can recommend Hotel Le Regent in Paris.. great location, very helpful desk staff ( made calls for me and printed out stuff when I lost my purse ) .. not sure if they have triples though.. you may need to email and ask if they would put in a cot ( how old is child... ?)

Triples and Quads are harder to find, rooms generally smaller than american hotels , and June is super busy. Good luck.

Found a triple room @ Hotel 3 Poussins in the 9th arr using Agoda (don't any experience using this site so looked them up and they get poor customer ratings).

Another option is a charming hotel called Le Chat Noir! in the 18th arr

Thank you!

I have stayed near Le Chat Noir at Hotel Terrass but I think you made the better choice for traveling with a preteen girl. Le Chat Noir is right in the middle of a bunch of x-rated spots. Take a look around on Google maps. I have stayed near there but the 6-minute walk makes a huge difference. One of my favorite restaurants in Paris is nearby but the specific location of Le Chat Noir is not ideal for your family IMO.