Thanks for your reply Tacit. It took me a while to find this since I don't log in very often nowadays.
I found a way around this particular conundrum, and it worked to my surprise. Actually, I was very surprised indeed, looking at the software home site, which says you cannot do this.
I deleted the pre-installed installation on the Mac I would rather use it on, which refused to work, and emptied trash. (I also deleted the preference file.)
A couple of days later I reinstalled it and supplied the 24-digit key and the 9-digit registration code which I have since I bought the license. It worked.
Blimey. The app in question is very rigorous. But what the hey, it worked.