I'm sure that the word weir as used today almost always means a dam. When I started this thread, I was thinking about the song Salley Gardens, which has the lines

She bade me take love easy,
as grass grows on the weirs.
But I was young and foolish,
and now am filled with tears.

Now this is an old song, and maybe in its time, weirs were also levees. Maybe...

However, have you ever tried to compose a conventional song, one which makes sense and rhymes at the same time? It is not easy. I don't blame the author for seizing upon a little-known meaning for weir in order to find something that rhymes with "tears."

My Sing-out book says that this song came about when W.B. Yeats attempted to reconstuct an old song from 3 lines he remembered an old peasant woman singing in the village of Ballisodare.

JTT: you may be right about the water plants that cling to the weirs. I don't think country people would call it grass, because it doesn't feed the cows, but maybe, when one needs a rhyme badly, things like that can be overlooked.