The transformer is probably at neutral ground already
The primary of the transformer will be at whatever potential the incoming neutral is. I think in most countries in the world it's required that neutral is very close to earth/GND potential, but consult local code on this and measure if you want to be sure. The phase will evidently cycle around this by up to (+/-) 325V in the UK and Europe (230VAC).
The secondary of the transformer will be floating by definition, unless it's fixed at some potential, which is usually not the case; see below. The fact that the secondary of a transformer floats is actually a good thing from a safety viewpoint.
If you use the ground wire there is a possibility of creating a ground loop if the wall socket happened to be wired incorrectly.
No. Safety ground is not connected to the electrical circuit. It's connected to metal parts of the chassis and/or enclosure. Connecting safety ground to the electrical circuit will trip the earth leakage protection. Hence, the connection of a safety ground cannot and will not ever create a ground loop, at least in equipment made by anyone who knows what they're doing. It's a safety feature.
The iron core of the transformer is often connected to safety GND/earth, but the core has no electrical connection to the windings.
AFAIK this has a plastic housing, so technically a safety ground is not even required, even by modern standards. You can connect a two-prong connector to this. If you open up the unit, you'll probably see that the GND doesn't go anywhere anyway, at least nothing particularly meaningful; maybe the xformer core; see note above.
Has anyone any experience of doing this and lived to tell the tale?
I've owned and repaired a fair number of Durst transformers and I have yet to come across one with a 3-prong cable. They all come with the standard 2-prong connectors that were standard decades ago.