Anne, sorry today hasn't gone as planned. 
Feel free to ignore, I'm sure you've thought of these ideas, or they don't work for you. The Q's are just thoughts, you don't have to answer.
It's not just academics, especially as she's struggling socially and emotionally. This is where the school should be focusing. DS' are academically able, but DS1&3 have EHCPs for SEMH and ASD respectively. As a recently diagnosed teen girl there must be something to make life easier for her, and you, before her MH deteriorates further.
The school should be dealing with bullying. Are they helping with social skills? If DD can't manage DT or PE they could focus on these when in student support. She's not left to her own devices in student support is she? What about ELSA? Is student support open at lunchtime? Do the school have a lunch club for those struggling at lunchtime? Have they asked the specialist teacher for advice or to observe? Does DD have a key worker/mentor? Does she have a time out card? Have seating plans in all lessons got her sat in the best place for her (front/back/side/near door/near window/on own/next to friend)? If DD doesn't have to mask as much in other lessons DT and PE may be easier. What are her organisational skills like? SALT/OT/EP assessments?
Do another group in DD's year do the tech she is doing now? If so, DD could stay where she is and join the next group doing that rotation. Alternatively, what is student support like? Some are suitable to do some form of food tech, some even have microwaves or ovens, even if it isn't the dishes the class are doing. Gloves can help if there's sensory problems of food on her hands. School could provide 1:1 support if she did make it into the classroom.
PE - do they have a gym? I know 2 pupils that don't do PE but are supervised in the school's gym. Does DD like any of activities done in PE? If so, as other groups rotate around different activities she could join the next group doing the activity she's OK with. Being unable to tolerate the changing rooms can be sorted by changing separately. Do others in the school have physiotherapy when she has PE? If so, dependent on the type of physio, I have seen it work where a student joins another's physio session. Do they have a multi sports group at lunch for targeted pupils? Is it only participating in the activity that is the issue or all of it? Could she umpire/coach/video?
Does DD like horses/horse riding? The RDA are brilliant, and I know pupils who go to the stables the afternoon/morning they have PE.
Life skills lesson #1 began yesterday evening. Teeth brushing. Oh dear. This is going to be a long process. Normally DS' have a go with constant nagging then DH or I finish off to make sure they are done properly. Last night and this morning they had a go then used disclosing tablets and finished off themselves. It took an age, lots of instruction and I still ended up finishing off DS1's.
Apologies for the long post.