Looking for advice for climbing San gorgonio with a small child

My 5 year old son and I have taken on the 6 pack peak challenge together and we have currently completed Wilson and baldy. I'm trying to plan the rest of our hikes throughout the summer and I'm a little concerned about doing San Gorgonio in one day. He is not a normal 5 year old, he has amazing endurance and seeks adventure constantly. He's no stranger to long treks either. We did The long trail to mount Wilson in one day and he has also climbed half dome while hiking in with a 10 pound pack. My concern is that the San Gorgonio hike is long and super tough on the Vivian trail and I'm wondering if there are places to camp along that trail so that we can do the hike to the summit in 2 days. And if there are camp sites along that trail, how do I get an overnight permit to camp? I've attempted to secure a permit for Vivian trail but all I see are Day hike permits available. Can anyone please help us?