I am teaching high school right now. I love it. High risk youth in inner city school. Lots of gang activity. These kids know most of everything in the book on causing trouble. Unfortunately for them I wrote that book 30-40 years ago. One shooting this year at school (kid got shot in the leg) and one other gun incident, but it was just intimidation.
I love it because I look for the kids having troubles, or the ones being bullied and befriend them. I am considered a “cool” teacher and kids will skip class to come to mine. Its very hard for a bullied kid who has no advocate at school - I try to be that advocate. One of my favorite kids is a girl, literally born to a crack prostitute, in a foster home with 24 other kids, has had a h-a-r-d life - but is so smiley and positive. She is a bit awkward and needy, and this bothers some of the other kids who have needs of their own. I have subtly tried to help other kids see what a wonderful person she is - and I feel it has had some influence.
However regarding homeschooling - all the real work that is done at a public school could be done at home in 3-4 hours max if there is a bit of focus. Half the day is wasted with announcements, getting out materials, interruptions from some jackass, etc. That leaves alot more free time in their day.
That free time can be spent in various social activities at church, Barnes and Noble book clubs - they meet like every Wednesday and discuss a book they all read together, gardening clubs, athletics, Girl scouts,hiking and adventure clubs, theatre, etc. There are tons of free or low cost activities for youth - plenty of ways to develop socially. Being bullied is not socializing. Some activities like a book club may attract more adults, but if she is as good a reader and mature as you say she might really like it - and you can go with her - great time spent together.
But I would give her the choice. We always planned on home schooling. When my oldest was ready for kindergarten all of a sudden I decided; to ask him what he wanted to do - after all even though he is only 5 its still his life. He said he wanted to go to public school, but he wanted a walkie talkie and that way his mom could stay out in the hall and they could communicate by walkie talkies while she peeked in the door. He was dissappointed to find out we would not do the walkie talkie thing. But he chose the public school to my dissappointment. But public school has been great for him, he is a junior in high school now and taking college level calculus math
My son had some bullying problems - we bought some books. There is a series of books on bullying - very eye opening.