Problem solved, legally!
While reading the Ts&Cs of the iTunes store agreement, I came upon this:
a. Scope of License: Licensor grants to you a nontransferable license to use the Licensed Application on any Apple-branded products that you own or control and as permitted by the Usage Rules. [Emphasis added]
Since "any" would include both Macs I now own, I signed into the iTunes store from my second Mac, and saw that "Install" was available for Affinity Photo. I clicked on "install" and, lo and behold, version 1.5.2 replaced the old version, and I now have 1.5.2 on both computers.
My bad for not having realized this before.