The OP was Federation vs Peacekeepers. If it's all of Trek vs all of Scape, Trek wins because it has quite a few different omnipotent aliens running around, let alone a gazillion different plots and super weapons, especially if you add in the books and comics.
As far as Feds vs Peacekeepers, that's probably more interesting than most here would admit. The Peacekeeper Command ships are sizable - various different stories on the size, but the lower boundary is about 1500m, much larger than any comparable Trek ship. They do have shields, very effective ones -they just can't fire out of the shields. But that offers some tactical advantages. The crew size is 50,000, and it carries numerous smaller vessels, including Prowlers and Marauders, and is escorted by Pantak Vigilantes. And while Frag cannons are their man armament, they also use Pulse weapons that are an energy/plasma based weapon. And the Vigilantes carry an immobilizer pulse - an EMP weapon very similar to the Breen weapons that completely disabled the Fed ships for a long period in the Dominion War and were responsible for the destruction of the Defiant.
I'd give the Feds many advantages, including ship speed, weapon range, overall tech level, and shield technology. However the Peacekeepers have both far larger ships and more attrition units in the vast fleets of fighters and attack bombers, the EMP weapon, and massive troop numbers compared to Starfleet. The difference for the Feds would be their crew. But in a space battle it would depend on the nature of the engagement, and a Peacekeeper Carrier group could easily be construed as a serious threat to a small squadron of Federation ships.
While the Federation has a greater diversity than the Peacekeepers, including some worlds whose inhabitants have unique abilities like the Vulcans, Deltans and Betazoids, the Peacekeepers have several allied or conquered races. These include the enslaved Pilots and Leviathans, the enslaved Banik, the Colata Hunters, alliances with the Hynerian Dominar, and the enslavement of the Delvian home world. Other allies and minions include Vorcarian Blood Trackers and Tragan Mercenaries.
As far as Kirk vs Chricton, Chricton knows and idealizes Kirk, almost as much as Kirk does himself. If it came to it, Kirk would be able to use that legend against him.
