I've standardized on TF4 for a long time. The alkalinity seems to work better at removing any residual antihalation dye stain, although complete removal occurs during the washing stage. Such residual dye might not be apparent on a pyro stained neg, but still has some effect. But it's in color masking work that it's least welcome; and I don't want to be bothered with different fixers for different projects. TF4 is also my standard print fixer, and saves a lot of time and fuss there.
An alkaline afterbath in relation to pyro stain is redundant, as others have already explained. It just increases overall fbf staining, and is not proportional like development staining. My attitude at this point is, If it ain't broke, don't try to fix it. In other words, TF4 works well for me. But it would be interesting to hear experiences about TF5 too.
You don't need a hardener with pyro negs. It's a tanning agent, self-hardening. But I have never found the need of a supplemental hardener with any kind of film. And with respect to stop baths, with certain films very sensitive to this like Efke 25, I employed stops so weak that it might as well have been plain water. Made no difference with respect to the pyro effect.