Géméllus is by far my highest recommendation. It’s surely great for dinner, too, but we have not yet made it there for dinner since the lunch value is such a steal.
When you go to Les Parisiens, make sure to order the fish mains. They are much better than the others. Desserts are also usually great.
I second beam’s suggestion about Pierre Sang Oberkampf (slightly better than Gambey IMO), and we usually go there on Sundays ourselves. This one place you should go for dinner, not lunch, for the full experience. Tip: Do not get the wine pairings here; not great value and not particularly interesting wines. Order a bottle of bubbles instead. Tip 2: you need to pay in advance online or they may cancel your reservation if you don’t reconfirm a few days before.
While by no means the only place to get it, we usually do steak at the very touristy, no reservations Relais de l’Entrecôte in the 6e. It’s just about the only touristy place we do go, but we enjoy the experience and the food.
We very much liked Akabeko for lunch, but I do agree with beam if one is only going to do one “special” lunch, I’d choose Alliance as well.
Cafe des Ministeres is very rich, somewhat heavy traditional food, so make sure that’s what you want if you go (it is indeed very good at it). It’s also not that easy to book last I checked, so you’ll want to be online to make the reservations right when they open them up.
I fully support your idea of sandwiches at/from Le Petit Vendôme.
Panurge in 2e is about a half hour walk from your hotel, so not sure you’d count that as walkable, but I’d recommend that too.
After Géméllus, the place we most enjoyed last year - not walkable - is Brion in the 9e.
I hope you and your family enjoy your trip!
As it happens, we will be arriving in Paris on April 3rd ourselves this year, also staying in the 11e.
