Hello there.
Well, you sound as though you have A LOT on your plate, and your feelings of being overwhelmed are not irrational.
As far as the BPD goes: I've heard that it can take a while to find the right medication for that. Would a diary help? a. it gets feeling outside of you b. you can chart your moods and see if there is a pattern - which might indicate that "tweaking" your medication is needed?
Have you read the thread on mental health and becoming a mother? It might strike a chord or two. There is a lot of stress in mothering, especially when the children are young, and it is to do with the psychological shake-up of dealing with someone so intimate, small, vulnerable, and so on AND the sheer gruelling, non-stopness of it. And the lack of support that seems to be normalised in our culture.
I don't know where you live but I'm in London and a lot of qualified teachers - with children - work as T.A.s. It's clearly not ideal if you need teaching experience but ... they are doing it because it works a bit better with their lives than full-on teaching. So acknowledge the pressure you are putting yourself under. Sowornout has a point. But it is your decision.
If you really are desperate for teaching experience, perhaps a sideways route? Agency work? But ... that might all be there two years hence, and you might be in better, less exhausted, place to go and grab those opportunities.
Whatever you choose, please do take a moment to pause, mentally give yourself a hug, and try and make a decision based on taking care of yourself, and being kind to yourself. You're a mother - try and mother yourself. Imagine you're many years older - what would you want your daughter to do, if this were her, and she was asking you advice?
I do think that one thing you might consider is acknowledging that you are feeling overwhelmed because you are overwhelmed. It's not you, it's your situation. I may be so wrong here, but I do know that sometimes it has helped me to begin to find a way through when someone has explained to me that I am finding things hard not because I'm a failure, and it is not only hard in my imagination, but because the situation is genuinely tough. Somehow, accepting that the fault isn;t mine seems to take a load off me, and I am then able to assess the situation a little more clearly.
Good luck, love. Well done on getting that PGCE. And total respect for finding the energy to keep churning out those job applications. Look after yourself, my dear.