Wow! You are busy! You are so blessed to have 5 kids. I only have 3.
I feel for you.......I can see why you are contemplating sending one of your kids to public school.
One thing you may want to consider is will your 4 year old feel left out? If I'm understanding this correctly you'll have your then 8, 7, 3, and 2 year old at home?
Is there any way you can lighten your load this year? combine history? do less science?
Next year will be a whole new ballgame. Your then 8 year old may be doing more work independently. Your then 2 year may not be clinging to your side. .....who knows?
Every family is different. If this is what you need, then this is what you'll have to do. I think it'll all work out. :)
Good luck. This is supposed to be a fun journey not a stressful one. :)