Well, it has to be McAlpine's, because that's tye name of the firm. I suppose Dominic might have sung McAldine because someone was worried about libel or something.
I've always heard of it as Dominic. I think it's quite likely that some other fella might have come up with the nickname "McAlpine's Fusiliers", and that Dominic took it from there. There's a Darky Finn in the song.
Someone ask Frank Harte. If anyone knows, he knows.