Really, we need separate discussions dealing with the unrelated song and tune that (partly) share a name.

The Princess Royal tune was attributed to Carolan by O'Farrell, c.1810, and by Bunting (on what grounds I don't know; he noted a version from the harper O'Neill in 1800) in 1840. No reference to any Carolan involvement that pre-dates the 19th century, it seems. Bunting reckoned that the "Princess Royal" was a daughter of Macdermott Roe, but this seems to be anecdotal at best. The tune has appeared as Miss MacDermott, but this appears to be derived from Bunting, so proves nothing in itself.

The truth is that we can't say for sure, beyond observing that the attribution is unproven; and that this is a case where people have tended to select evidence to support a proposition they want to be true rather than examining all of it and drawing dispassionate conclusions. The tune was in print in England by c.1730 in Walsh's Compleat Country Dancing-Master, as 'The Princess Royal, the new way'. That, too, doesn't prove anything in itself, but the fact that it was some three-quarters of a century before any attribution to Carolan can be found does explain the circumspection of editors like Moffat, who, while making clear their personal feeling that the tune was Irish, were too scrupulous to claim on the evidence available that it definitely was. Frank Kidson (who Moffat quotes; they were friends and collaborators) was in the opposite camp: his piece from Groves can be seen at Musical Traditions:  The Arethusa.

See The Fiddler's Companion for extensive précied notes, with the caveat that the sources are not all reliable and the summary sometimes too condensed to tell exactly who said what. In particular, nothing said by the discredited maverick Grattan Flood should be believed unless confirmed by independent sources. The title really doesn't represent any difficulty even if the tune was composed (or adapted; possible antecedents have been proposed) by Carolan or any other Irish musician; it would be anachronistic to imagine that an Irish musician of the time would feel unable to name a piece for a member of the British aristocracy, though plenty of people who ought to know better have made that assumption.

One last thing. Princess Royal, the new way implies that there was an "old way", and this may have been the Princess Royal, which appeared in the middle of the previous century. Claude Simpson (The British Broadside Ballad and Its Music, 582-3) gives that tune as printed in c.1662, and mentions that two ballad operas of c.1729 and 1733 specify Princess Royal for songs, but that these don't match the stanza pattern. Perhaps the "new" tune was intended. Without knowing what the stanza pattern actually was, we can't draw conclusions; but such vaguary is typical of the whole argument, which will probably never be resolved.