I took A. dracunculus to mean either French or Russian, since some references seem to give dracunculoides as a synonym. Trying to sort that out gets complicated and it doesn't really matter to me which it is, unless there's a difference for this use as opposed to cooking. I'm guessing if it's tarragon it's Russian given the use to which it is being put, and I think it has paler foliage than the French?
The garden is about an hour and a half away so I'm not sure if we'll be going again before the flower season ends. If it's a perennial it'll still be there though if they're doing the same design next year but I have another idea - there's a herb garden not far from here so maybe I'll head out there at the weekend and check out their tarragon and see if it looks the same. Thanks!