I have heard it sung as:

And the wild mountain thyme
All around us is perfuming.

The "blooming heather" bit is still there in the chorus. This means that both the verse and the chorus rhyme. I think it's quite likely that at some stage the oral tradition got the two tangled up and the "perfuming" line got lost along the way.

I can't think of any reason why a song that is slow, or a verse that consists of a single sentence, shouldn't rhyme. The rhythm scans, and the melody leads the ear to expect a rhyme. The other verses rhyme. Why not go for it and (re-)introduce the rhyme?

Marje