I am not sure the cheap=carbon steel connection is that self-evident.
Thread chasers for Coventry die heads are meant for production work, where people are making money. The conditions under which they are used in an industrial setting are such that a carbon steel item would likely be unsuitable (not long enough life) or uneconomic (cannot cut fast enough to make money on the part). I think it would be quite difficult to find a die chaser that is not (at least) HSS.
Set up a holder in your QC toolpost as if it is to hold a boring bar. Clamp the chaser into that.
Hey everyone, today I did the thing! Thanks everyone for your input, it turned out great.
Here's a video of it in action:
It's not perfect, but it works and I can feel the extra power (calculated at about 2.75x higher). I can always remake it in future now that I have the tools and technique.