My advice, just stop arguing with them. It’s not worth it. I’ve run into this a fair amount, and I’ll usually make a fair effort to point out that their supposed qualification is irrelevant, it’s a logical fallacy, and they’d be better off finding a more solid way to defend their point because it’s not going to convince anyone. Unfortunately, in my experience, most people who argue like that aren’t really interested in discussing the idea, they’re interested in either being seen as more intelligent or just having their opinion validated.
Hell, in my own experience, I’ve had people try to trump my opinion with qualifications, unaware that my qualifications are superior. Unsurprisingly, when they drop theirs and I trump them, there’s not a deference to my opinion, just an argument about either why mine aren’t actually superior or how mine are irrelevant. For instance, one of my favorites was when a coworker still working on his Associates in Information System Management was making ridiculous assertions about programming and had forgotten that I had my Masters and was working on my PhD in Computer Science (not just more advanced, but also more relevant), and when I mentioned that, he started throwing out stuff about how that somehow meant he had more knowledge about it, pretty much by spewing utter nonsense. When it got to that, I pretty much just rolled my eyes, laughed on the inside, and just let it go.
Around the same time, I had someone else questioning me on a topic that was directly related to thesis research, and his qualifications were “I worked with computers for 20 years.” Even direct cites from the patent on the technology in question, were just met with “well, I don’t think it works that way”. So, again, I just let it go.
As I see it, I have an intellectual obligation to point out a fallacy and try to correct ignorance or misinformation, but once it becomes clear that the person I’m talking to is not discussing the topic in good faith, it’s just time to let it go. Sure, they may be able to convince themselves they’ve gotten the better of you, but this is also someone who has an ignorant opinion in other matters and can’t discuss them fairly. So, why should you even care what they might think?