We don't know the true limiting factors in TOS style starship fights, but both TOS and the TOS movies had our heroes engage only individual enemy ships in phaser fights. Engagements against multiple enemies did not include exchanges of fire, but rather a desperate attempt to avoid such ("The Enterprise
Incident", "The Deadly Years", the simulation in ST2).
On the other hand, the ability to target a small and maneuvering enemy existed in TOS already; the hero ship was at a disadvantage only when deprived of warp maneuverability, which somehow affected targeting, or otherwise suffering from significantly lower speed than that of the opponent ("Elaan of Troyius", "Journey to Babel"). Given a properly working warp drive, the ship could maneuver extensively (including "pivoting") and defeat a foe with phasers or even with a point-and-shoot volley of torpedoes.
This would suggest that Kirk's avoidance of multiship fights was not due to his ship being unable to target multiple opponents, but rather due to multiple opponents possessing too much firepower for the hero ship's shields to withstand. Then again, in the M-5 simulation against multiple strong opponents, the rapid, accurate and tactically well thought out switching of fire from target to target seemed to be the main advantage provided by the new computer. Perhaps the crucial issue would be avoidance of enemy fire and finding the right openings for one's own volleys, using elegant maneuvering and timing, rather than anything as brutally technological as the raw targeting power of one's sensors and guns?
The supposedly added number of phaser emitters in the refit might not affect the outcome much, then. Coverage blind spots might be eliminated, and perhaps phasers and shields would gain in absolute power or even in relative power against their Klingon equivalents, but battles would still be decided on the two sides maneuvering cleverly to avoid being hit. This would be much like WWI fights where even a single hit might cripple you or your opponent - and completely different from TNG battles where, Horatio Hornblower style, the side that withstood incessant and unavoidable point-blank pounding longer would triumph.
Applying this to the original question, I'd think Kirk's ship would perform much the same before and after the refit in terms of capital ship adversaries: nobody in the 'hood would dare challenge her one on one, but a formation of three would still guarantee victory to the opponent, and presumably 2:1 odds would be good enough for an attack already even if not a surefire way to defeat Kirk.
It's too bad we never saw Kirk take on a large number of small opponents either before or after the refit (this happens in several novels, but never on screen).