Welcome

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I'm sorry that things have been so difficult for you. Outlandish accusations can be so hard to understand. It can make you feel crazy trying to figure out how they came to their conclusions.
Why do you need to get him to believe you? You know your truth. You know that you did not do the things he is accusing you of so do you really need to try to convince him otherwise?
There is something more going on with his anger over these seemingly little issues. It could be he is feeling out of control. He could be worried about something or he could be feeling frustrated that he had to be inconvenienced. One thing that might help is to
validate what his feeling, worry, concern, is over even addressing his accusations of lying. The hard part can be figuring out what he is REALLY upset about, not about what he is saying.
For instance about the door being unlocked, completely ignore the accusation and instead of asking him "Why would I do that?" You might say something like, "It's worrisome thinking someone could break into the house easily with the lock being unlatched. Does that worry you too?" Keep asking him and talking to him about what he is feeling because for someone with BPD feelings=facts. And you can't begin to talk about facts until after the feelings are dealt with. It's important that when he makes those accusations, that you
Don't JADE . When you JADE it can cause pwBPD to feel like you are just making excuses or telling them that what they are saying doesn't matter. Does that make sense?