Have they offered an occupational health assessment? You've not been there long so technically they can fire you without reason and I can see how they might think "personality clash" and decide its better to let you go now than maybe be dealing with it for a long period of time. If you want to argue unfair dismissal from a disability point of view, you may be best raising a formal grievance now about their decision to make you work with JK and the lack of supports in place. You would need to be able to evidence (eg through GPs report, occupational health support etc) that you had a formal diagnosis and that there's something about this person that impacts you "clinically" ie that your response is beyond your control and therefore you need some form of adjustment.
Otherwise, could you use this time to really work on coping mechanisms, getting them strengthened so that you're better able to cope with the situation? Maybe slow down on the EMDR (I'm not sure yet more exposure to trauma is a good idea when you're still dealing with a situation that you find triggering and while you have inadequate coping mechanisms). It feels a bit like you're pinning your hopes on EMDR being the thing that makes it all ok but in reality it may just make things a little more manageable. My own experience is that I've needed a variety of different supports and therapies over the years and have also had to do a tonne of work myself.
I think your employer is being realistic in giving you a point at which they need to see some change - I'm guessing in their eyes you've been unwell for most of the time you've worked there and it could go on indefinitely. While they have a duty to make reasonable adjustment, they may argue that what you're asking for isn't reasonable in the kind of workplace they operate and so you're not fit for the role, which would be dismissal on capability grounds, not because you're not technically able but because the role impacts your health so much and to a point that they can't support you.
Hard as it is, at least you know what you're working with, you just need to decide now how to go forward.