Jeff, your idea of the Garden Wall-Granite Park-Swiftcurrent Pass is a good one. Yes, the Granite Park CG is reserved for through hikers only (although they opened it up this past summer to anyone, due to the heavy snowpack in the park, to serve the disappointed hikers with other reservations that had to be cancelled). You could always book a couple of rooms at the Chalet to make it work. However, the ~3000' steep descent from the pass down to the Swiftcurrent Lodge is a real knee-buster, and maybe not suitable for the younger kids.
1) You could do a 7-8 mile in-and-out to one of the lakes on the west side--Kintla, Bowman, or Logging. Not the most spectacular hikes in the Park, but flat, easy hiking, and scenic campsites once you get there. It's mostly hiking in the forest.
2) This past summer I did a 2-night hike to check out 2 campsites I'd never stayed at--No Name Lake and Upper Two Medicine Lake. Fairly easy stuff, and even more so if you shorten the hike by taking the tour boat from the east end of Two Medicine Lake to the west end (saves maybe 2 miles). If you do this, stay at No Name first, as it's a bit less scenic than the beautiful Upper 2 Med campsite.
3) A very easy 3 mile hike from the Goat Haunt boat dock to Kootenai Lakes (where you are guaranteed to see some moose). You have to drive to Waterton in Canada, then take the tour boat to Goat Haunt. Might be a bit cumbersome for you, logistically. Passports are probably needed to cross the border.
4) Several nights can be had in the Belly River country. The trailhead is near the border crossing, but still on the U.S. side. It's about 7 miles to the Gable Creek campground, and from there there are two choices--to Elizabeth Lake, or follow the trail toward Stoney Indian Pass, where there are several stellar campsites on Cosley, Glenns, and Mokowanis Lakes.
5) Another fun in-and-out one nighter is to hike to Gunsight Lake.
6) A 2-night suggestion would be the Quartz Lake loop, which begins and ends at Bowman Lake on the west side of the Park.
7) A very easy overnighter can also be had by hiking from the Fish Creek CG 4-5 miles along the west side of L. McDonald to the campsite. Not terribly scenic until you actually get to the tent site, but then you get quite the vantage point.
One thing I might add is that the Many Glacier area is full of grizzly bears. The three valleys that converge at the Many Glacier lodge (Grinnell, Swiftcurrent, and Iceberg-Ptarmigan) are intermittently closed due to bear activity. You should carry 2-3 cans of pepper spray wherever you hike in GNP, and make lots of noise. Good luck with your planning. You really can't go wrong in GNP.