Hi Tojian, welcome to the group

Glad you reached out for some support as you wait to add therapy as well. 12 weeks is way too long to wait; I hope the list speeds up for you.
In the meantime, you're in a good place to connect with members who have gone through situations like yours. Lots of people here have had to come up with non-intuitive plans for "who gets what and when and how" after a breakup with a pwBPD (person with BPD), as, like you mentioned:
No matter how much I tried to show her with facts on paper she will never see that this was how it was.
For "broadly normal" breakups, people can generally sort out who gets what mostly fairly. For breaking up with a pwBPD, nothing is ever enough, and "fair" means "she gets everything". I can relate (my husband's kids' mom has many BPD traits, and after they divorced and split up stuff, she -- months afterwards -- demanded he give her back furniture that they had already agreed to divide...)
Now my problem at this moment in time is that she has left the home for a month now, left two dogs that she wanted in case one didn’t love her and is living with her mum. She claims she has now become homeless again and is expecting me to give her all the things bought since we lived together.
Are they her dogs? Yours? Unclear? Does she want the dogs back, or just the stuff? Dogs are trickier, so I'll wait to hear more from you before we brainstorm.
Sometimes, some pwBPD will make seemingly irrational choices. They might say verbally or in a text "I want my stuff back", but when you compare their actions to their words, it seems like they drag their feet, prolong getting the stuff, and don't do the actions you'd expect. Some pwBPD do that in order to maintain (or believe they are maintaining) a connection with or control over you, especially if their stuff is in your home.
Given that you're posting on the Detaching board, it sounds like you are committed to being done with the relationship?
One strategy that some members use is:
Rent a cheap storage unit for a set period of time (some people do 6 months). Move all her stuff in there, and either physically give her the key, or send the key registered mail (signature required), with a letter saying the time frame that you will pay. Then, instead of her making it your responsibility to "be at home when I say so, be around when I get my things, be available to argue over whose is whose", it is her responsibility to use the key to get her stuff.
You are also allowed to change the locks on your house (I think I'm tracking with you that it's your house, not co-owned/co-rented?).
Sometimes, when a pwBPD is involved, it "feels mean" to do things that are actually normal boundaries. If we can get past the "feeling mean" or "feeling like we are not helpful", then we can move forward with our lives, without taking responsibility for choices and actions that aren't our responsibilities.
Let us know your thoughts, whenever works for you -- and again, welcome;
kells76