Since the term "atomic matter pile" can be used as an archaic term for a nuclear reactor, I have to agree that the USS Republic had one. And since there's a reference to "points" which have decayed to lead in WNMHGB it's reasonable to assume that they used fission for minor energy generating purposes on board the Enterprise, too.
Nuclear reactors can mean either fission or fusion although I'm thinking Tenacity is on to something. "Atomic Pile"
without the word "matter" is the archaic term for nuclear reactor (includes fission and fusion). So an "Atomic Matter Pile" must mean something else and the existence of an "Antimatter Pile" would suggest that to be the case.
Perhaps both the matter and antimatter are processed through some heating reactors (piles) to get them to the proper temperature before intermix? Or maybe "matter pile" and "antimatter pile" mean the same thing and are shorthand for "matter-antimatter pile"?
From "Charlie X"
CHARLIE: Yes. There was a warped baffle-plate on the shield of their
energy pile. I made it go away. It would've blown up anyway. Well they weren't nice to me! They wanted to get rid of me. They don't now.
From "Court Martial"
KIRK: It's no secret. We were assigned to the same ship some years later. I relieved him on watch once and found a circuit open to the
atomic matter piles that should've been closed. Another five minutes, it could have blown up the ship.
From "Journey to Babel"
MCCOY: Sure. A formal reception tonight, a hundred and fourteen delegates aboard for two weeks, thirty two of them ambassadors, half of them mad at the other half, and the whole lot touchier than a raw
antimatter pile over this Coridan question.
From "Spock's Brain"
SCOTT: Either a
nuclear pile a hundred miles across or...
Additional info regarding "lead" being an indicator of nuclear fission power... in "The Paradise Syndrome" we hear Scotty complain about Spock burning out the star drive with the line, "That Vulcan won't be satisfied till these panels are a puddle of lead." This could further suggest that there are processes with the star drive (matter-antimatter engines) that could decay or melt something ("points"?) into lead and it isn't exclusively tied to nuclear fission... YMMV.