I would just take out the child who is unhappy and not worry about anything else for now. Either the others will continue to go to school and be happy, or they will want to be home ed and they can come out and be happy.
I know it doesn't feel that simple! I took my younger daughter out at age 8 and my eldest was quite happy in secondary. I'd mooted HE to my eldest a couple of years beforehand when she'd been bullied at primary and she'd refused point blank.
For a year I had my younger daughter out of school and my elder daughter in school. DD2 got all my attention in the day and therefore I could devote my attention to DD1 in the evenings and both seemed happy with this arrangement. Home educating just one child was lovely, easy, peaceful, and a lot of fun.
As the year progressed, DD1 started to think more and more that DD2 was onto a good thing. She would come home and hear that DD2 had been to the Open air museum or the Roman palace or to the adventure play place with her friends and started to think DD2 was having a lot of fun whilst she was stuck in school. DD1 is over 3 years older than DD2 and bright but despite this she could see DD2 somehow catching her up - understanding maths DD1 was struggling with, reading the same books, and within a year at home DD2 had caught her up on the piano as she had so much more time to devote to it. We weren't happy with some of the secondary teaching and the more dissatisfied with school she got the more she seemed to struggle with friendships too. By the end of Y7 she was begging to come out of school.
This was not what I had bargained for. I had a nice calm life that worked, had got over the panic of taking DD2 out, and had no idea whatsoever how to educate a 12 year old. But I couldn't say yes to one and no to the other and risk a lifetime of resentment on my hands. So we took her out, promising to home ed her for 2 years only until GCSEs.
Well, she stayed at home until 16, got 10 good GCSEs at home, mainly at A and A*, and is now in her second year of college doing A levels. The girls are now brilliant friends and have a really strong sibling relationship. Given that they are now 17 and 14, I have a relationship with them that many parents can only dream of with their teens. We are all so close and get on brilliantly. My bond with DD1 is incredibly strong. Yet both girls have their own friends and very happy social lives. DD1 is at a friend's house this weekend and DD2 is off out later on.
I can't guarantee you won't have all three at home eventually if you take one out. Of course, your eldest may well stay happy in school. If you follow their lead they will never have cause to resent you. DD1 says she would have resented it if I'd taken her out of school when I took DD2. What I can tell you is I was at least as scared as you both times (people who have been on here longer than me will vouch for that!). Despite it being the most difficult choice I ever made, it was definitely the best thing I ever did. Good luck in your decision making. 