The CA should win this fight most of the time, assuming her captain handles his ship properly.
Incidentally, the USN actually studied a somewhat similar problem and gamed out a series of engagements between a CA & and ACR during the 1920s, both of which are detailed in Friedman’s book on US cruisers. After the WNT was signed, the USN looked at the possibility of modernizing its remaining armored cruisers in 2 separate studies, as they were not only effectively exempted from the treaty, but were bigger, more heavily armored, and in the case of the 3 surviving Tennessees, carried bigger guns than any cruiser legal to build under the treaty.
The second series of studies included an analysis of the modernized ACRs retaining their original main battery [based on BuOrd's assumption that replacing them with the 8"/55 of the treaty cruisers was impractical; the 8"/45 of the Californias could be increased to 30 degrees giving a max range of 28k yards; the 10"/40 of the Tennessees could be increased to 40 degrees, giving a range of 31k yards] albeit with increased elevation, pitted against Pensacola.
According to the resulting calculations, the 10" of the Tennessees could penetrate Pensacola’s side armor at any range it could hit, and the deck from 18k yards out, Pensacola would actually be immune to the California’s 8"/45 between 15-18k yards. On the other hand, Pensacola’s 8"/55 , as calculated, not only could penetrate the decks of the ACRs [the proposals studies included no changes to their side or deck armor] from 18k yards out, and the side armor out to 20k (a bit optimistically; revised penetration tables based on service data suggest about 17-18k), but also enjoyed a range advantage, being capable of firing out to 35k yards. Nor were the 6" guns the armored cruisers could also bring to bear considered to offset the 6 extra heavy barrels Pensacola had.
Operational studies suggested that even with new engines, assuming they increased the speed as promised, [the second studies proposed replacing the engines of the ACRs with the powerplant used on the carrier Ranger, which was expected to increase their speed to ~26 kts] the reconstructed ships would be too slow to operate independently, or with the carriers or new cruisers. They would have to be used in squadron strength screening the battle line or convoys against enemy cruisers, and supporting destroyer attacks carried out in the face of a cruiser screen in order to be effective.
These results, combined with concerns over what weapons the cruisers might be up against at the end of their post-modernization lives in the 1940s when they already weren't up to facing existing weapons, and the political issues inherent trying to rebuild 25-30 year old ships after a difficult political fight to buy new cruisers for the first time in over a decade, half of which were intended to replace the ACRs in question scuttled the modernization plans.
Back to the scenario in the OP, assuming that the County-class CA in question has a belt & is loaded with full-tilt AP shells in the magazines, then the engagement should go in a way similar to the studies of the modernized ACRs vs Pensacola.