When we're doing a 'fire with a bunch of non-singers present, we usually start with stuff "everybody" at least remembers little bits of, like
Down in the Valley
Home On the Range
Long Long Trail
When Johnny Comes Marching Home
You Are My Sunshine
If they "work", we'll do some other "camp" type songs, like
The Deacon Went Down
You're in the Army Now
that are easy to learn and to make up verses for once you get the hang of it.
If they don't work, we pretty much let the non-singers listen, and the "singers" present do whatever comes to mind. Usually, no matter what, there's a few people who won't join in.
If I understand correctly, you're looking for songs for repeated sessions, and "non-singers" who attend frequently may learn more new (to them) songs. We are doing a "one time" thing for many of the campers. We have a new group every week. Some campers stay two weeks, and many were there the previous year(s), but we never know what's going to happen. We tried printing up songsheets, but its DARK out there, and they didn't help much.
We ALWAYS end with Amazing Grace.