Well, I've never been there for labor day, but last year I was there at the end of July in 2007 and I was the only non PCT thru hiker that I saw backpacking in the park for the 5 day trip I did. Now I did see several parties in the Caribou Wilderness to the East where I came in and left. I suspect that if you avoid the camp grounds, most people will be day hikers and not backpackers.
There isn't really a difficult hike in Lassen if you are talking about elevation changes. You set the difficulty by increasing the distance. The toughest climb I encountered was the climb up to Willow Lake coming from Butte Lake, but that was more due to the steepness of the trail.
My recommendation would be something that visits the Painted dunes and Cinder Cone. The most obvious would be a loop around Snag Lake and starting/ending your trip at Butte Lake. You can increase the distance if you want something more difficult by taking in some of the other lakes in the park.
If you truely want solitude, the most eastern part of the park (Willow lake etc) doesn't get many visitors and the trails there show it. The trail south of the junction to the Caribou wilderness to Red Cinder Cone, you often have to follow the markers on the trees since the trail disappears in places. A causal hiker would probably get lost there since you can't always see the next marker from the current one; though following what seems to be the faint trail and hope usually will put you at the next marker in a minute or 2 of walking. I lost the trail once and spent 10 minutes trying to find it again; I was more careful after that.