Presently, it appears to be a cosmetic change. However, since it was given it's own stats in the gamemaster, and using &costume in the search bar doesn't list it as one. The "no form registered" in Pokédex is another thing to take note of as well, as you described.
It's presently a unique situation: either an alternate form with different stats, with the form not being recorded in Pokédex, or a cosmetic change with separate stats from the original form.
I can't help but to think that Niantic messed up in terms of programming/coding it as one or the other, Wouldn't be their first time messing up after all.
Well in any case, I'm keeping extras *in case it could be transferred*, even though I don't expect it to be able to.
I mean, some Pokémon don't retain their form changes after battle (Castform, Cherrim for example), and I don't expect those alternate forms for those to be transferable too, but without precedent (using Let's Go transfers) as an example, it's hard to know which scenario is correct for sure...