I am going to send you a big hug because I am dealing with a similar toddler issue, but my oldest children are in a public school. One of DH's arguements against me bringing the kids home is that he is worried this toddler will be regulated to being penned in by me while I teach my older kids.
Some of the questions going though my head after reading this thread are:
Do you have the extra-income to hire outside help or pay for an outside day? If you do so, rather than a daycare MDO situation, are there options for things like tumbling classes or active classes that don't require the parent there? Sometimes the problem with my little one is that he has to burn a TON of energy for about two hours before he will settle to play. It took a lot of trial and error to figure that out.
Is corralling him in a room a feasible option, and would that calm him down or make him feel ignored and therefore act out more?
Would you consider enrolling the older kids in school for the Spring semester so you could focus on 1 on 1 time with this child? Maybe you could get him into a routine where he 'does school' with mommy (just one on one time) and then you could pull the olders back out next summer and sort of roll them into your new routine with the young one and a new recharged you.
I am sending a hug because it is so hard when you have this active type of child. Having mom at her wits end and stressed and upset and unhappy is not healthy for your family. Is it worse than sending them to a school? I honestly don't know. The school my children currently attend is fairly mild and has it's annoyances, but we don't have gangs or big trouble makers. I still do afterschool activities with my older kids. I didn't call it afterschooling before I found TWTM, I called it around-the-clock learning and lifelong learning. During the day, I do fun craft activities with my little ones and take toddlerzilla to the park or now that it's colder, to a tumbling class.
I wish you peace in your choice. I know that being a primarily homeschooling site, suggesting a temporary enrollment in school might be a faux-pas, but there are many of us that supplement and use homeschooling materials while kids are in school temporarily or as a choice. 6 months of school might be a life saver for your sanity and thought of as an outside temp class, it might be the solution to your problems and give you time to focus on your youngest and for supplementing your olders, or it might be a confirmation that you are truly a homeschooling family and you just need a bit of time to refigure your approach.