I think you first and foremost need to completely LET GO of the idea that this man will ever be reasonable and concerned with the accuracy and truthfulness of events.
I found that once I let go of any expectation of reasonableness - it was easier for me to stop rising to the bait. I expected him to be unreasonable, he was unreasonable as I expected so there was nothing to see there or to react to.
At the moment, your emotions and brain are still telling you that if you just argue back with the truth, he will realise the discrepancy and change. He will not.
This is his narrative and he lives in that narrative and in fact believes it to be true. Forget about trying to convince him otherwise.
Just use the stock answers as suggested by PP.
Re-direct to the actual topic that needs discussing.
Pretend you are talking to a brick wall - because that is what it feels like - and just repeat again and again - “ok, so what are we doing re: DC next week?”
If you try to “correct” his inaccuracies you are fuelling the conversation.
Learn to stop giving a shit about what he thinks - especially if you are sure you are right - and focus on getting the information that you need out of the conversation eg what time are you picking DC up tomorrow? That’s
all you need - the time and place. Everything else is noise.
Also stop caring / fearing what he says about you to others. I’ve been divorced over a year now and I am astounded at how many of my friends/family who have only now said “we always knew he was a …..”.
Astounded because we were together 23 years and no one said anything to me prior to the divorce and now friends / family are using words like “manipulative”, “overbearing”, “talked over you”…. So that means people will be able to see him for who he is.
As long as you remain focused on your co-parenting goals, everything else will come out in the wash I think.