As long as you’re careful while the emulsion is wet you should be okay with non-hardening fixer. It only has an effect until the emulsion dries anyway.
That said, old-fashioned Kodak hardening fixer is cheap as dirt and it won’t hurt anything if you use it, so why not? You’ve got a lot of old film to use it up on.
I use Heico Perma Wash, which performs the same function as Kodak Hypo Clearing Agent, with both traditional, hardening sodium thiosulfate and rapid, non-hardening ammonium thiosulfate fixers. As I understand it, it helps to make the thiosulfate more soluble, reducing washing times.