Is saying "...so I know how science works." likely to convince people?

[Dr H]

I trained in the "hard" sciences and technology (physics; electrical engineering) and worked in that arena for some years.  Later on, I acquired training in so-called "soft" sciences (developmental and educational psychology) and more recently have worked in that arena.  
Oddly enough, the most consistently useful training I have had is in philosophy, because it taught me to bullshit convincingly, which, properly applied,  I have found to be an asset in most arenas.

A great many non-scientists who think  they "know how science works", really have no idea.  One dead giveaway is the statement "science proves . . ." followed by pretty much any positive assertion.  The only thing science ever proves, is to prove something false.  In the pursuit of  falsifiability one may amass evidence in support of an hypothesis -- but the hypothesis is never established in an absolute sense; it must remain potentially falsifiable, else it is not a valid scientific hypothesis.  A lot of people find this concept very hard to grasp, or if they do grasp it, very hard to accept.

As you know, it can by significantly more difficult to set up properly controlled, blinded experiments in the social sciences than the "hard" sciences.   For starters, there are almost always more uncontrolled (and unknown!) variables when working with people and human interactions.  And you can't just isolate a person into a petri dish or a cloud chamber for six months to see what happens.  Even setting up a control group study means getting an acceptable human research protocol, with voluntary informed consent, etc., through IRBs and ethics committees -- no mean feat.  And one reason why so many social science studies use comparison groups, rather than control groups.

Still, it can be done; it is done; and meaningful results do emerge.

So, what do you think, is saying "I've published three papers about linguistics in peer-reviewed journals (about...), so I can safely tell you that's not how science works." more likely to be productive or counter-productive?

It's certainly relevant -- it establishes your credentials prior to pontificating in your field of expertise. 
Whether that's productive or not, is another question, and probably depends on context.