One day in Paris with Kids
I like le Reminet but the last time I was there with a kid it was an unmitigated disaster.
The menu is interesting but very short. My friend's kid has an even shorter palette, which tragically does not coincide with the menu.
In addition, the wait which was really not long was too long for that child.
In short, did not go well.
Several of the adults had to apologize to the restaurnt afterwards.
The restaurant was very gracious about it. However, one can expect Le Reminet to deliver gracious service. One can't expect it to speed up the wait and cater to a child who really only eats one food subgroup.
I would recommend Dans Les Landes. The tapas format is fun for adults and children alike. It also suits a party with people of varying appetite. One can order a long series of little plates, and then share mains, as we did. Not every resto in Paris has this flexible kind of offering.
Dans les Landes is not walkable from the places the OP is visiting but can be reached easily by metro line 7, station Monge.
Pare down your itinerary. With transit time to and from Gare du Nord, and waiting time at these places, there is no way you will be able to effectively see them all (assuming you actually want to go inside the Louve and up the tower).
I'd say go straight to Eiffel first thing, stll be prepared for a very tiring wait. Afterwards, it would be just a short walk over to Violon d'Ingres for lunch (1 star) now open 7 days a week, with reasonably priced, good cooking, reasonably casual.
Plan on 1.5 to 2 hours to really enjoy this lunch, three courses each plus modest wine plus coffee plan on about 275E, double that If you happen to go with the Jules Verne suggestion and probably more like 2.5 to 3 hours there.
Afterwards line up at the Louvre and prepare for another lengthy wait, you might see one or two galleries before they start closing up. See what I mean, you just do not have enough time to cram all this in, pick and choose your attractions wisely.