I haven't been to Avalanche Lake, so I can't speak to the camping there, although I think I've read of people camping there. I wish we'd gone to see the lake just to scope out camping potential for future trips though! (it was already a long day for us so we didn't make the slight detour to see the lake). If you camp there, you'll have to let me know how it is.
As you drop East of Trail Rider pass, there is a phenomenal campsite before you get to Snowmass Lake (see attached pics). If my memory is right, it was maybe half a mile above Snowmass Lake. It has some tree protection and amazing views to the lake below. Just look to the left of the trail as you descend and you'll see the use trail with the campsite basically right off the trail. There's no water there, so you'd need to pack in enough water to dry camp. This is why it's often available, though---people who don't know about it see the awesome campsite but aren't prepared water-wise, so they end up going all the way down to Snowmass Lake instead. Snowmass Lake is extremely crowded, but there are a lot of campsite opportunities in the area. You could also go a little past Snowmass Lake as if you're headed towards buckskin pass (maybe a mile from Snowmass Lake I think) and there are good camping opportunities in that area as well I think (I've never camped there personally though). That would add less than a mile of hiking to your trip. I have never been on the rest of your loop (down the Maroon Snowmass Trail and then the West Snowmass Trail), so I don't know about the campsite availability there either.
If you don't have a trail map for the area yet, I would recommend the Sky Terrain map instead of the Trails Illustrated map. It shows the user trail that I referred to before that will eliminate the walk to the Lead King road.
Also, there are a ton of cows along the Capitol Creek trail, so you'll want to get water further in (higher up) than the cows. Also, we came across a herd of around 200 sheep way up high (probably at 11k ft or so) right above a stream that we had just filtered from (and there was sheep manure all over on the other side of the stream, that we didn't see until after we had guzzled water and crossed the stream). My point being, you'll want to filter/purify water up there even if the streams look good.