Hi there @EK
I may be a little late to the game, but for future reference, I am a big fan of Big Creek Campground.
This could easily be a 3 night adventure by camping the first night at the main campground. Then there are a few backcountry sites (37, 36, 38) while taking the 2-night Mt. Sterling Loop Trail. Overall it is a 17 mile loop, with a bit of elevation gain when climbing up to Mt. Sterling, which may not thrill the littlest (or you having to carry littlest). Just make sure to get permits through NPS.
Backcountry Campsite 37 is the closest to Big Creek Campground at 5.5miles, with 36 just a few trots further.
THREE DAY LOOP
The first day will have the most activities and stops. Along the Big Creek Trail you will find Midnight Hole, great for swimming, and Mouse Falls. The first backcountry campsites (37 & 36) are along the river and relatively flat, a good area for a family.
The next day is mostly a wooded climb up to the Mt. Sterling fire Tower, 4.4 miles on a steady up. Beautiful, but no great stops for smalls kids on your way to backcountry campsite 38. Camping that night is not as flat, but there are some great little alcoves for family time. Before you head up, make sure you've got plenty of water... the water resource at this site is a bit further than the campsite (then a climb back up).
The last day is a 6.6 mile up & downhill as you make your way back to Big Creek Campground. Wooded and lush, but no waterfalls or bouldering.
OUT & BACK TRIP
It may be fun and easy to try making a basecamp by booking at Big Creek Campground. Stay there the first night, then backpack up to backcountry site 37. With plenty of stops along the way, and such a flat & friendly backcountry site, it's a good start to testing how your littlest does. Stay the night, and then hike back past all the beautiful waterfalls and swimming holes again. You end right back at your Big Creek Basecamp.
EXPORE COSBY CAMPGROUND
Another option, that I have not personally done, is looking into an out and back from Cosby Campground. Start there and hike the Lower Mount Cammerer Trail to Backcountry Camp 35, the next day head to the Davenport Gap shelter, and then reverse it all.
Be sure to plan ahead and speak with the Campground about overnight parking & policies so there are no unpleasant surprises.
Enjoy the Smokies!