This is an odd request, but does anyone know of a place that can accommodate a picnic-style gathering on a Saturday early afternoon? Perhaps a cafe where you can bring your own food, or a casual wine bar where you can order meats and cheeses?
Within a month or two someone asked chowhound Paris about restaurants that would close for a special birthday party and got some suggestions. You might try these places and explain your "special circumstances" and see what can be done.
A few Christmases ago my daughter flew home with a fresh salmon my mother-in-law didn't want cooked in the house. I called a local restaurant we then frequented and asked them to cook the salmon for us. They did and we bought some sides and drinks for the family...eight of us. They didn't close for only eight. The mgr wouldn't take anything for cooking or serving the salmon, so I tipped everyone in sight. In the Adirondacks, we (The Hotel Saranac) regularly cooked trout caught by guests and served them in the dining room.
Your request is unusual but not unheard of.
"A cafe where you can bring your own food". LOL. I never tried it in Paris, but the USA Starbucks certainly don't care about bringing in anything as long as you buy a drink from them. There are a couple dozen Starbucks in Paris. As far as anywhere else in Paris, not a recommended move, you will stir up a lot of ire! (And be thrown out)
Or, just go to one of the beautiful Paris parks that are ubiquitous all over the city and have a real pique nique!