I just watched the video, and want to point out that he might not be lying. There's two things I can see (one I've actually seen), that would cause him to be bad while not lying about his belt, which would also explain the defensiveness.
There are some schools that add BJJ to their curriculum, despite the instructor not actually knowing it/knowing it well. So a TKD or WC school can claim to also teach BJJ, teach it really badly, but also rank people in it. And don't get found out as long as they don't have open mats or go to tournaments. In that case, he wouldn't be lying, he would have just been duped. And he wouldn't want to tell you his instructor if he realized it at some point and is embarrassed he got duped.
The second thing is there are sites that teach online, and give rank out. The most well known only gives online rank to blue belt, but I would be very surprised if there were not ones that give up to purple or higher as well, in BJJ. So he could have gotten his rank without ever actually rolling. That would explain why he claimed to be a purple belt, but felt uncomfortable rolling, and got defensive about his instructors (he'd have to admit it was all online).
It's also entirely possible he lied for any number of stupid reasons-to brag, he wanted to have something in common with you, his friends told him to, he likes lying, etc. But that's not the only plausible reason. The only way you'll know is by talking with him about it, not asking on a forum.
And if you genuinely want the answer, ask in a way that doesn't make him feel attacked. Which I'm going to go off on a limb and claim wasn't the case here, if you kept pressuring him to roll, had other people watch your roll to test him out, took a video of it, and then started posting the video here. Which, regardless of whether he got a rank he shouldn't have, or was just lying early on and got stuck in the lie, would make some prideful/stubborn people double down in response.