Has the etymology of "Ballygrant" been given?

In 2000, "Philippa" reported that the macaronic lines in the ca1840 broadside read, "I wish I had you in Carrickfergus/ Agus ní fada ón áit sin baile cuain."

As Philippa acknowledged, the translation is not 100% reliable because the broadside prints the Irish lines phonetically.

Neil Comer then translated the "baile cuain" as "Quiet" or "Harbor Town."

"Harbor Town" sounds as though it would fit "Ballygrant." But is that really the etymology of the town's name?

The point is that the phonetic spelling on the broadside, as reported here, doesn't capitalize "Baile Cuain" as though it were a place name. If Ballygrant were intended, I'd expect to see an "r" in there somewhere. Of course, the (monolingual?) printer could have accidentally omitted it if there'd been one, just as he didn't think to capitalize.

I'm not rejecting the probability that Ballygrant was really meant. I'm just pointing out a detail that might or might not bear on the matter.