I have had children at school and have also home educated. Right now your child needs to feel secure and the right conditions to learn in.
My concern about forcing her into school is that it's a long day for a child who is hypersensitive to stimuli and I think she might be too overwhelmed to learn which would create a vicious cycle.
I also do not think that socialisation is your biggest problem right now. My home ed kids went back to school with high confidence levels and were popular because they played with everybody and didn't feel confined to cliques. They were also academically comfortably ahead which is so easy to achieve with one child and your child is clearly bright.
The psychiatrist is going to recommend school because there may be support there and home ed is an unknown. I would consider it and probably ignore it just for now. If this was an adult, I think they'd end up being signed off with with stress if you continued sending them into work in this state, regardless of what the original problem was. Given the nature of the illness let's give the girl a break.
She's still little. She has lots of time to develop and grow. She can do after school activities where there will be other children and even if you only know one other home ed family, that can be enough for now. You might be surprised at how many there are.
I would explain to her that you understand she doesn't feel relaxed at school just yet and she has some extra learning to do. You have listened to her when she has told you that she wants to learn at home. You are willing to help her learn at home so that she is doing the same work as the other children when the time is right for her to return.
This will not make the separation anxiety worse. It's there already. She is likely to outgrow it as she grows in confidence and develops naturally. I would nurture every relationship you can with others. Maybe there's an older lady who she can play with and have a biscuit for half an hour. That's a start. Maybe there's a family at the park she will play with, provided she doesn't feel the need to constantly check you haven't wondered off.
I don't think there is a quick fix here. The language is a problem. Can you speak to her in Portuguese? I would see if there is a lovely student who could come and play with her a few times a week on a regular basis in a kind of mother's help role. Another chance to speak the language and another opportunity to find out she can manage without you. Don't force it. Just be very boring and make sure the helper is relaxed and has fun stuff - stickers, slime, a magazine, a craft. She can't talk in English to your daughter unless absolutely necessary and then she must explain what she's said in Portuguese and check understanding later.
An online teacher can be another way of having contact with the outside world.
Get her into every activity you can afford and she will try. Private lesson at first are often easier. Physical things like gymnastics, riding, dance and swimming might be absorbing and less tiring than learning an instrument.
What you need right now is a relaxed child who is able to learn. School is not able to provide that. She has regressed and it might just be too noisy and too demanding. It sounds miserable for her. You have time. Relax. Focus on getting her ready.
I think you might be very surprised by how differently a staged return to school goes in a year when she has benefited from recovery and support, is not daunted by the work and has had some time to develop.
But what a tricky time for you all. I have been in a not dissimilar situation and I really do think that accepting where she is at right now is the way forward. And go outside lots.
A bit random but daughter really benefited from having a Maine Coon cat. I refused to touch it so that cat knew who it belonged to. They are incredibly clingy which, in a strange way, helped my daughter to understand that all the anxiety around separation was unnecessary and a little bit funny. The cat sat outside the door complaining if she went to the bathroom. My daughter would reassure her but the cat wouldn't be told. It sat on the kitchen table and studied her as she ate her cereal in the mornings. It accompanied her on walks. It slept on her bed. It sat on her reading book and swiped her pencils off the table. It was a warm, breathing, huge companion and it gave her confidence.