I like the Independence Pass suggestion,
Unnamed 13500 is a fun, mellow hike with a trail up to the summit boulders -- but the summit hike would be 1400' gain. Possibly worth it for an out-and-back that may or may not include the summit -- still great views, marmots, snowfields, etc.
If your car can handle it, and it's something you'd enjoy, you could consider driving up Mosquito Pass, and hike up Mt. Evans B. That's 'only' 400' gain, but is above 13k feet. There isn't a maintained Class 1 trail, but there are bare spots among easy talus. The radio towers/antennas are ugly, but might be an interesting distraction for the kids -- and the views in both directions are fantastic. Here, you'd get views of the Sawatch, but would still be just East of them...and cool mining buildings to check out on the Leadville side.
Cottonwood Pass, and
Ptarmigan Lake/Jones Mt., are just a bit longer and higher than you mentioned, but might be worth a look (the lake and saddle could be a good shorter hike).
So neither of the above are in the 400-800' gain/Sawatch/12-12,500' suggestions you were looking for.
As for Pikes, which is even farther outside the parameters -- I (like many) love the Barr trail, but disdain the summit and the road. I'm afraid (and I've seen kids comment on this) that hiking to the summit is perplexing -- "Why didn't we just drive?" and "Are all mountains like this on top?" -- so they get a confused view of summits. Then again, there are donuts!