Answering after doing a bit of research prompted by a comment by Waruyama.
Referring to these as Reuleaux polygons, e.g. Reuleaux triangle, might get you somewhere. These polygons are much closer in appearance, to my eyes, than polygons with rounded corners (which are, to me, quite distinct and not a sufficient description of these shapes at all). However, the term has a number of issues:
It is not well known outside geometry and specific technical fields (they’re used in some engines, for example), and the name doesn’t hint at anything.
Reuleaux polygons are very specific mathematical shapes with particular properties. You cannot simply take a polygon, curve the sides a bit, and claim it is a Reuleaux polygon—that would refer only to a polygon with very specific curves to the sides.
Only polygons with an odd number of corners can properly be called Reuleaux polygons. So a squircle cannot be a Reuleaux polygon, no matter how carefully you curve the sides.
And for that matter, those corners are sharp, not rounded. Though saying “Reuleaux polygon with rounded corners” might get you around that one.
Finally, it appears that there is a company called Reuleaux that sells paraphernalia for vaping, and that tends to dominate search results, which will cause problems in understanding and discoverability.
Reading the linked Wikipedia page does offer a link to circular triangle, however, and that term has much greater promise: it is a general term for triangles formed from circular curves. The Reuleaux triangle is one, but this term can also cover a variety of other shapes. In fact, it can cover shapes that we would not consider the same as your “trircle,” since the curves that form it can be convex or concave. In these figures they are all convex—which can be communicated, per the article, with “convex circular triangle.”
Since we are also not being very picky about our curves—they are not necessarily circular curves, really—we can generalize that term too. AAGD’s answer suggests “convex elliptical triangle” where an ellipse is a more general term for curves that include circles, so that is a step in the right direction, but then we also really aren’t necessarily referring to elliptical curves either (and this can also run into some confusion with elliptic geometry, which again look similar but are not quite these shapes).
So I am going to suggest that we could use the term “convex-curve triangles,” and more generally, “convex-curve polygons.” Probably “with rounded corners.” That would cover precisely the shapes in question.
It’s also basically unheard of. Google finds 6 results for
"convex curve triangle". One is selling jewelry with stones cut into the appropriate shape, and another appears to be an art gallery with a geometric bent, and both are using the term to refer to the “trircle,” so at least we aren’t contradicting what little prior usage there is, but that’s not saying much.
"convex curve polygon" gets 10 results, but they all appear to be highly-technical geometry research papers.
Finally, I would note that the term that was most accurate for these shapes while still being within the realm of “people actually use this term” was “circular polygons,” from which we can clearly see the actual derivation of squircle: square-circle became squircle. Likewise, triangle-circle becoming trircle, pentagon-circle becoming pentircle or pentarcle or something, and so on. So while these names are not frequently used, as noted in the question, they are both accurate (as shortenings of the “circular polygon” terms) and a clear extension of the better-known “squircle.” So my conclusion, ultimately, is to echo filip’s answer, and suggest that these names are the best choice for regular use.