Many café terraces are warmed -- and they will let you bring food outside of meal hours. On sunny winter days, a picnic in a parc is always possible -- so much so that you'll have to go early to have a seat on the sunny benches. I'm not sure to understand the concpet of picnic inside -- you mean like "is there a big public hall where I could picnic?".
Oh boy, this is the most interesting question I've seen on food sites.
I'd invest in a candle to light at Notre Dame for the one warm week that always happens sometime.
But pavillions in parks are pretty much as cold inside as outside. Why not go to Rosa Bonheur in the Buttes Chaumont where you can get some picnic type stuff (sardines, charcuterie, etc) to the Left of the Bar and I'll bet no one will notice you've also brought along a ton of herring, salmon, oysters, wine, Pierre Herme, Chaudon chocolate, etc, too.