Definitely step back with regards to the Gsces and emphasise to him that you don't mind if he fails some of the time, really don't mind, what matters to you is that he feels better longterm.
I mentioned VITAMIN D3. Fresh air is important but if he hasn't been getting much of it (you mentioned this was something you were trying to encourage) he may be Vitamin D3 deficient. My son was, and it made things worse for him, much worse. GO OUT TODAY AND BUY FROM THE SUPERMARKET OR PHARMACY A VIT D3 Supplement for 1,000iu tablets (25mcg) and give him 4 today and one a day for the next few weeks. Or failing that, if you can fit it in, get him a blood test for everything, folates, vit d3, iron. You may find there is a more physical reason for his lassitude and anxiety.
You are doing brilliantly really you are, the only thing that remains now is to care less about the exams and more about his health, long term, and that may mean giving up on this particular school pathway, and following plan b in the short term. It may seem to you that it is obvious is he just did x y z he would feel so much better, but if you have ever had PND you will know that it is not so simple as getting out in the fresh air, having some exercise and keeping on top of the washing up,and following a few simple routines. Grownups with mental health issues, whether it is PND or something else feel poleaxed by it. It is not just support and counselling that get them through sometimes it is drugs, sometime it is TIME, sometimes it is thinking out of the box about the way you live.
Could you take some time off from your job, for you, not him. Signed off by the GP with stress, or parental leave perhaps? It might be worth the financial investment now, for later. Goggle threads from parents with children with anxiety and you will see it is not as simple as just feeling angry with selfish behaviouir, although that can be a very important part of acknowledging how devastasting it is for us as parents, and how much we have to think out of the box to cope with this "behaviour" from our children. Coping might mean taking time off work.