Even with 6 kids, I try really hard to keep things from becoming too chaotic, because I can't deal with tons of noise and chaos. So I sympathize with your son. If it's reached the point where he's sleeping in the laundry room and going to work with dad, then he needs some changes.
I would start with the room and general day-to-day functioning, because I think it's a huge deal for a tween if they don't feel like they have a quiet place they can retreat to. Is there any way to rearrange the rooms so that he has his own space? I understand if they need to share a room, but sometimes there are ways to reorganize the room so that the mess doesn't spill over onto his side. It helped my current 7th grader a lot when we just physically rearranged the room so that he had a corner that was clearly his own space (that his little brother is not allowed to touch). We also bought a new nightstand for his bed so that he had a space where he could put his things that is not shared with his brother. But I have to be on top of the little guy to be sure he isn't touching big brother's things and to be sure that he cleans up his stuff each day. It isn't fair to 7th grade son if I'm not monitoring and teaching the little guy. I know that's hard with a baby, but it helps to build it into the daily routine.
Do you have a good daily routine going? I don't know how old your baby is, but I would work at building a solid routine around the baby's naps. If it's chaotic when you work with the younger kids, I would plan for oldest ds to work independently in another room during that time. When our curriculum hasn't been conducive to working independently, I have had older kids practice instruments or do assigned reading in a different room while I work with littler kids. But I think it helps a lot for middle schoolers to have a few subjects they can do independently and a daily checklist to work from. It's a lot of work for me to make a checklist, but it really helps the day go smoothly (though I would try to keep the number of items short at first so it's not overwhelming). We've done daily checklists in 5th & 6th and then transitioned to keeping a planner in 7th & 8th.
I also don't think there's anything wrong with moving to Calvert or something else more structured & independent if you think it would help. Some kids thrive with structure, and it's okay to get that structure from an outside provider if that's not your natural style.