The analogy I'm getting for your situation is asking the question "what tool do you need to build a house"?
The answer isn't just "a hammer". It's: "lots of tools, at different times, and sometimes the same tool a few times, and sometimes multiple tools for the same task".
I think I'm reading a couple of things going on with the circular conversations that you two are having.
One is that it's good you're recognizing that
you have 100% control over your participation.
We're allowed to think about our own values and priorities, and decide for ourselves what rules we'll have in our own lives, that are totally under our control. A shorter way to say that is that we are allowed to have
true boundaries 
A person can have any kind of boundary about circular conversations, ranging from "I leave the room as soon as I think one is starting" through "I choose to engage with the conversation as long as it stays calm" through "I give circular conversations 5 minutes of my time and then I move on". What's important is that you respect
your own boundary, whatever it is. For you, it sounds like you are deciding not to participate in interactions where the other person is lying about you, to you.
It'll be critical to remember that the boundary is for your own protection,
not for controlling or changing anyone else's behavior. Your boundary protects
you from hearing hurtful and untrue things about you. Will he keep saying those things after you hang up the phone? Will he keep believing them? Will he say them to others? Who knows... he might? But the important thing is that you protect yourself from hearing it, no matter what he does.
...
The other thing I'm picking up on is the idea of validation (another tool, along with boundaries).
There's this "pop culture" idea that validation means agreeing with whatever the other person says, no matter how crazy. Then we find ourselves saying "He can't expect me to validate that, it's a lie!" or "How can I validate the crazy stuff she thinks is true?"
That isn't quite it, fortunately! At BPDfamily.com, we don't endorse agreeing with stuff that isn't true, or just saying "yes, dear", or placating the pwBPD. None of those are true validation.
True emotional validation is a next-level skill that involves pausing, putting ourselves in our loved one's shoes, thinking about how we would feel if we really believed what they were saying/thinking/feeling, and "hitting the validation target". It's not easy or intuitive -- but it gets both parties out of the conflict trap where the pwBPD is expressing feelings in a low-skill way when making factually incorrect statements, and the non-BPD is getting defensive or wanting to "set the record straight" -- just continuing the conflict cycle.
I think I'm seeing a glimmer of the feelings behind his words, here:
he’s justified in excluding me from a recent family event because I “flew with my son to (an English city) 4 or 5 times in the last few years including this January leaving him at home to mind my dog”.
Obviously, it isn't true that you live a jet-setting lifestyle where you can leave him on the porch and waltz around the world on airplanes every couple of months.
If he has BPD, then he has significantly impaired skills when it comes to managing and expressing his feelings, which may tend towards feeling victimized/abandoned. He won't be able to say things like "when you flew to [city] a few years ago, I struggled with feeling alone and left behind". That level of insight and vulnerability is deeply impaired for many pwBPD. What he is able to come up with is -- he feels so lonely and "left behind"... he doesn't have insight into why... what on earth could've made him feel so forgotten... his partner leaving multiple times is the only thing that could hit so hard... so it must have happened... and you said you went twice... so you must have gone more, if he feels it so strongly.
Sometimes, when we validate the feelings behind the words (note: we are not agreeing "it happened the way he said"... we are validating the emotions, not the "facts"), that can halt a circular conversation. Sometimes circular conversations happen because the pwBPD is feeling unheard or misunderstood about their feelings.
One approach when hearing something off the rocker like "you went to [city] five times and left me alone at home" would be to find the emotions behind those words, reflect on how you would feel if you felt that way, and engage with that:
Him: you always leave me alone at home when you fly around all the time.
You: oh wow, babe... that would hurt to feel lonely and left behind like that.
Because if I
felt like my H was always leaving me behind when he went on trips, I would feel pretty lonely, too.
...
Neither boundaries nor validation is "the one tool" you'll ever need to build the house. They're two pretty good tools, and there are a lot more that can help make your relationship more livable for you. You may find that you need
multiple tools even in one conversation; for example, if after you offer emotional validation, he just keeps going with the circular conflict, you are allowed to exit the conversation: "hey babe, I need to take the bread out of the oven, I'll be back in 30 minutes". No justification necessary, just you taking care of yourself
...
It's good you're able to exit in-person conversations when you need to.
What makes the phone ones more difficult, would you say?
