There are two BLM campgrounds just outside of Atlantic City, itself right outside of South Pass City (but on the Atlantic side, get it?). Both are primitive with just water and vault toilets IIRC. But, both offer some shade and are fairly high in elevation, so reasonably cool all summer. Reasonably.
You may hook a left at Henry's Lake, ID on the way to Virginia City, cut through part of the most awesome Centennial Valley, then north to the Ruby River drainage to Cottonwood CG, with a great number of disbursed sites in the Ruby drainage. Leaving Cottonwood CG to the north brings you out at Alder, just up the road from VC by a few miles. Fuel up at Island Park if you are considering this 60-70 mile no-services gravel road route paralleling US 20/US 287/MT 287. Closer to the main road are NF CGs at Wade Lake and Cliff Lake, themselves maybe an hour's drive short of VC.
Up towards Granite/Philipsburg are a number of CGs around Georgetown Lake--nice but crowded and with lots of motorboat activity on this rare large piece of flat water. A NF CG lies along the headwaters of Flint Creek between the Lake and Philipsburg. Or you might consider jumping over to the west of Philipsburg into Rock Creek for a 41 mile, 3 to 4 hour stroll through a beautiful forested canyon alongside a blue ribbon trout stream. There are a good half dozen or more NF CGs along Rock Creek, with Bitterroot Flat, Harry's Flat, and The Dalles among my favorites. At the Dalles is a suspension footbridge crossing the creek accessing a small wilderness area and it's likely the kids would like that. The northern end of Rock Creek Rd dumps out on I-90 at Clinton, MT, maybe 20 miles west of Drummond and not far from the southern access to Garnet.
From Garnet back to Bannack, I'd take MT 1 back through Philipsburg to Anaconda, but there take MT 569 south over the Continental Divide to MT 43 west of Wise River. MT 569 isn't a superhighway, but it's all paved and it's totally scenic. We always seem to see a ton of wildlife between the Divide and MT 43--moose, mule deer, pronghorn, bighorn sheep, etc. Go east on MT 43 to Wise River, then south onto the Pioneer Mountains Scenic Byway. The Byway is around 60 miles of very smooth paved parkway going up to the headwaters of the Wise River, over a knockout plateau, past a free quartz crystal digging site (Crystal Park), past a rustic natural hot spring resort (Elkhorn Hot Spring), and back down the Grasshopper Creek drainage to MT 278 around 20 miles from Bannack. If you're considering Bannack during Bannack Days, I'd be checking on CG reservations pronto. I suspect they fill up quickly. If not camping there, it's a good 15-20 miles in most any direction to either NF CGs or disbursed camping areas.
If you route this way, and especially if you route to Lemhi Pass leaving Bannack as discussed below, you'll want to fuel in Anaconda and top off in Wise River. To the best of my knowledge, there is no fuel beyond Bannack until you reach either Salmon or Leadore, ID in that direction, Lima, MT over on I-15 south of Dillon, or Dillon itself.
Leaving Bannack for Silver City, look at routing south down Bannack Bench Road (graded gravel) for about 15 miles to MT 324, enjoy a few miles of pavement past Grant (no services) to the gravel road leading to Lemhi Pass, where Lewis and Clark first crossed the Divide. There is a nice kiosk with lots of historical info there. Descending the ID side via Warm Springs Rd (NOT Agency Creek Rd, which has warning signs a the pass advising of steepness), you pass directly by Sharkey's Hot Spring. Sharkey's is a very nice low-key BLM facility with the natural hot springs piped into two sizable concrete tubs/small pools, concrete aprons, nice changing rooms, and nice bathrooms. Nominal fee required. From Sharkey's it's an easy graded gravel descent to ID 28 south of Salmon, where you can either get back S --SE to I-15/ US 20 or go N to Salmon, then S along US 93 through the Salmon River canyon through Challis.
Though it's not a ghost town, you're passing right by Craters of the Moon NM at Arco, ID. There is a CG and the NM is pretty awesome with lava tubes as high as a house, cinder cones, spatter cones, etc. It appears as though the smoke just cleared from the eruptions, and in July when it's 100 deg F and you're surrounded by jet black basalt, it feels like it,too Ask me how I know.
Enjoy!
Foy