It's not 19th century, but I've never seen a kid who didn't like A Place in the Choir (click), by Bill Staines. Rounds are good, too. Gospel songs and spirituals might work, too.

Now, if you want 19th century, I love the songs of Stephen Foster and Henry Clay Work that you'll find at this site (click) - but most of those songs at least touch on the sad parts of life. "Grandfather's Clock" and "Long, Long Ago" (or Click here) are two of my favorites from the 19th century, but they don't fit your criteria. You didn't mention the ages of the kids - the songs in American Folk Songs for Children (Ruth Crawford Seeger, 1948 - found in most U.S. libraries) may appeal to kids under the age of about 10.
Another good resource is the Rise Up Singing songbook. I'm sure you'll find some good ones among the 1,200 songs in that book - but most of the songs in that book that are more upbeat come from the 20th century. Good luck in your search, Suzanne. Let us know what you come up with.
I have to say, though, that your restrictions make the task awful hard. I sing for kids all the time. If somebody asked me to follow the guidelines you set, I don't think I'd take the job.
-Joe Offer-