I have not engraved the bands for that clock, but have done some other dials, including dials like lathe feedscrew dials, which are orientated the same way, and I am familiar with the Merlin clock project. What's your specific question?
I have also dome some etching, although if I was making the bands I would engrave them by choice, as I think this gives a crisper finish to the sides and corners.
I would not make the bands the way that process is described in the book (page 50) but would either make the band from a strip and turn it, or, for preference, turn the band out of tube. I'm not sure if there is a convenient size of tube, but that would be the easiest way. Without removing the tube from the chuck, I would transfer the chuck to the CNC mill. Sorry if that's not your cup of tea, but it is the easiest way for me. That allows the engraving to follow the surface of the band, round the bend, so to speak. I have engraved plain flat dials mounted on a faceplate in the chuck, using a vee tool, but even if the bands remained in the chuck on the lathe, the problem would be the diagonal movements. Not impossible, I think, but very tricky to co-ordinate.
I believe a conventional manual engraving machine can engrave around a band, without much trouble, but I have no direct experience of that.
Marcus