Just me, but
I think that is unlikely to be a form of either C. peltata or C. nummularia.
In the photo of the UFO (unidentified freaking object), the blades are singular upon stems ---all of which isn't at all peltate to begin with: you can see the point of attachment at the edge of each discoid blade with a serrate edge.
In both C. peltata and especially C. nummularia, there is tendency towards several determinate branchlets, each leading peltately to a 'cup', all that blahblah coming from a single upright, rather than just individual blades-that-aren't-cups per upright that we see here.
Pauper that I am, I have not encountered this UFO before (Caulerpa is a fairly large Genus and as Dr. Guiry has pointed out, they are quite genetically variable), but I'm imagining it's either an undesirable hybrid, or else a variety ---both involving C. serrulata. Imagine a typical C. serrulata whose bades do NOT elongated, but are severely stunted to resemble discs.
I could be wrong (and I do have a perverse tendency to favor the longshot, oddball, alien-abduction guess), but if I'm not...
we've had sufficient hardship from varieties/forms of Caulerpa the past few years, so I STRONGLY advise caution on this thing's distribution until you can obtain a proper identification.
horge