For the sake of 14 months and potential £1200 (cost to do 4 gcses home schooled /online)
Bear in mind that 5 GCSEs A-C (or 9-5) including English and Maths is considered by colleges, universities and employers to be the basic minimum required. Obviously specific courses/more competitive universities may require more or certain subjects. It depends what your DD wants to do in the future but don't be so quick to narrow down her options at such an early stage in her life.
I'm speaking as someone who took my own DD out of school when she was 12. She had depression and anxiety, really low attendance and hardly any friends and was, like your DD, desperately unhappy. She says one of her biggest regrets in life is leaving school and she feels that her life has been harder because of this. She has just got a place at university but she is 20 and it has been a very long, hard road of emotional and academic struggle for her and it has cost me a fortune because doing GCSEs and A levels outside the school system is tough, or at least we found it tough. As Cornucopia55 said, you really are on your own once you deregister and we found it very hard to access help.
If she is 'on the mend' with therapy and medication, might it be worth trying to push through and finish school? Unfortunately in life there will always be difficult things to deal with and getting up every morning and maintaining decent attendance for work is a part of life that she will surely have to get used to at some point. Although it can seem like a good idea to pull your struggling child out of a place that causes them anxiety to give them a break from it all, I found that all this did for my DD was make her worse. She left the house less and less, didn't have many friends and her ability to deal with situations declined rapidly. School is tough for many people but so is work and life in general and school teaches you the extremely valuable skill of resilience.
I'm really not trying to scare you, I just thought I would give you the other side so you get a balanced argument. I think home ed can be a positive thing but you have to really work to make it that way - you have to put effort into making sure your DC don't become introverted and stay in the house all the time, you have to get them out and meeting up with other young people. It also costs a lot of money to complete GCSEs although there are ways to try and minimise the costs...is £1,400 the price for an online course? Bear in mind you will also have to pay fees to sit exams at a school as a private candidate, which can be high. Also if the subjects include coursework or practicals it gets very complicated and expensive. Best option may be to do IGCSEs, which have no coursework. It may be cheaper to buy the textbooks and make use of online resources, then just pay to sit the exams at a local school, rather than paying for a distance learning course.
Could you liaise with the school and discuss the possibility of going in part time or still being enrolled with the school to do the exams but agree that she doesn't need to go in? I never explored these options but I know someone who managed to arrange something like this. It might save you money on the exams. Talk to the school and the LA, see what is possible before you take the leap and cut yourself off completely.
Have you spoken to your DD about what exactly makes her anxious? Is it the school, the teachers, the students, the work? Does she have any other friends outside school?