Yes, Bingley, there is a lot of truth in what you have asked. The first quatrain, in particualr, ("Awake, for Morning into the bowl of night / Has cast the Stone that puts the stars to flight / and Lo! the Hunter of the East has caught / The Sultan's turret in a Noose of Light." {5th edn}{{EDIT: what AM I thinking about!! that is the 1st edition!}}) bears but a passing echo of some of the sentiments expressed by the Tentmaker. There is a vast difference in the words used to express the Dawning of the Day between the one I've quoted, which is his 1st edition, to the version he uses in the 5th edition, itself not at all the same as the 2nd edition!! Every quatrain is Fitzgerald's interpretation rather than his translation, of the original. This is shown, to some extent by the changes he makes to some of the quatrains from one edition to the next. (The absolutely top-famous quatrain about the book of verse, the loaf of bread, the flask of wine, and thou beside me in the wilderness, is a classic example of this - LIU!)
But I think - as BY has already suggested, that the strength of Fitzgeralds work lies in his capture of the spirit of Khayyam (Khayyam himself would, doubtless, have said it was the wine, not the spirit )

And to Avy, I will heartily agree that his chosen quatrain is also a firm favourite of mine - it was just, I think, that the previous postings had suggested the other two more strongly in my mind. And "The moving Finger writes ... " has been so overused - - -. (But is still a powerful piece of imagery, hey?)