Try lubing the stabilizers. I would just use dielectric grease on the contact points where the metal is coming into contact with plastic. Because filco uses costar stabilizers, they're easier to work on than Cherry ones. You can take the wires out and lube the wire and the plastic housing. If that doesn't do the trick you could also try putting a bit of foam on the inside of the offending cap to dampen the noise a bit. It's common to have rattle and ping on stabilized keys, but I've had good success with reducing or eliminating it. Another thing that might help is to do something to dampen the upstroke, like a QMX clip or zealencio. O-rings just dampen the downstroke. The topping-out sound on the upstroke is usually a clack, but the force of it may also contribute to rattle or ping I guess, and so a clip might help. But I bet a little grease will probably do the trick so try that first.
If you search for "costar stabilizer rattle" you will find tons of threads and some videos about different ways to address it.
It could be, as you say, that the acoustics are just different on the minila because of the layout. Assuming they're both using costar stabilizers, I'm not sure what they could do differently between the two keyboards for build quality that would have an effect on this phenomenon. Unless they're lubing the minila, but I doubt they are.
Rattle and ping drives me nuts too, especially where I've put o-rings or clips on the other keys so it stands out more. But in my experience it just takes some work and patience to get it to be quieter. I don't see it as a "build quality" issue, but maybe others with more experience have a different opinion.
EDIT: Here's a video showing how to lube the costar stabilizers:
