Perry-class frigates were designed as "jack of all trades, master of none" ships, to operate independently in low to medium threat environments . As such they were armed to repel attacks from all three axis (AAW, ASuW and ASW ) but never excelled in any of them.
Those kind of ships are excellent for Third World navy, with similar opponents, because they would be cost effective (1 ship instead of 3 ). But, I'm afraid they are not what Taiwan needs right now to deter China.
Currently, Taiwan has (and plans to build more) substantial number of ASuW missile boats. But those boats cannot operate far from shore because they lack air cover. Taiwan would need air-defense frigate or light destroyer with VLS just like they intended but never accomplished with Chen Kung class (Spanish Álvaro de Bazán-class frigate could be example of such ship ) . Perry-class with single arm Mk13 launcher is simply not good enough to go against PLAN any more.
As for ASW, IMHO in China vs Taiwan scenario that would not be so crucial for Taiwan. Simply saying, Taiwan would first need to break out into open sea to worry about Chinese subs imposing blockade to Taiwan.
Problem with buying powerful air defense ships is cost. They'll never be able to buy enough of them to become a serious threat in the power balance, and considering they will be constantly emitting in any real conflict, they will be among the first ships to be located and targeted.
Also, expensive air defense ships will siphon money away from other domains that could otherwise be more survivable, cost effective, and potentially cause the PLA to allocate more resources in dealing with -- submarines, specifically.
In that sense, smaller multirole frigates might be the best route for the Taiwanese surface navy at present while they try to fund other more asymmetric parts of their military.
But the power balance is only going to keep accelerating in its current direction, and the geography of the island itself doesn't do the ROC many favours in trying to fit a non-asymmetric war against the PLA.