OP this sounds really difficult.
I work in CS. I have a child with SEN. We are in the ECHP process. I find it immensely difficult to balance my children and work. I do have an understanding manager, which helps. I have been in CS for over 20 years.
Also fast tracked and until I was a parent I worked days off, evenings and was able to offer so much discretionary effort. I no longer do this and for a long time, I saw this as not being good at my job. However, I am good at my job, I deliver well and on time and manage as many projects as others. However, I also know when to say no, as I am working at capacity. I work differently but I am more effective than I used to be, partly because I have had to be. I have stopped going for promotions.
I have been off with stress and I don't believe I had any negative marks against me for this. I have managed large teams and have never been punitive towards someone who is off and has a genuine reason, which includes stress. Those I have had to manage through the sick processes either had patterns (always sick in school holidays) or poor attendance. They have also tended to be people with performance issues.
One of our benefits is sick pay. People don't like CS, and I don't want to derail this by going into public perception but this thread is full of it.
Your children need their Mum to be well, functioning and able to fight their corner.
Your department need you to be well, capable of doing your job and able to deliver what is in your job role.
Right now it sounds like you are struggling to do either. See the GP. It sounds like stress is what you need to be signed off for. Think about how you will fill your tank for when you return.
Look at the carers passport, this should help.
Either refer yourself or have your manager refer you for OH.
Have the counselling.
check out support groups for SEN, ECHP application both locally and on FB.
There may be difficult choices to make. We are about to downsize so I can afford to do less days. Not ideal, as the type of house we can afford to live in will make somethings tricker at home, but my daughter needs me to be free for her more. She is in school at the moment, but we suspect she won't be for much longer if we don't get the ECHP support in place. It is a hard choice, we won't have holidays etc, but it is the only way of surviving. Work gives me my sanity space, so I can only reduce not quit, unless I find another avenue to cope and keep well myself.
Look after yourself and whilst I wouldn't say my daughters needs have got easier or less - she is 9- I have accepted the changes which comes with being a SEN parent and adjusted what I want from life.
If is hard being a SEN parent. You can feel so unseen with friends, as it is hard to describe your experiences of parenting. I have lost friends, because I just can't work, be a good parent and see friends. I have tonight cancelled seeing some friends because my daughter had a rough start to the day, I was called into school and I needed to make up my hours this evening.
I hope sometime off helps you build your resilience