, what number am I looking for on the meter to indicate whether the 22uF/50V capacitor I'm about to connect to it is likely to blow up or not?
i never seen capacitor connected to rectifier blowing up
what you must keep your eye on is that the rectified DC may exceed the reading of the AC voltmeter attached to transformer secondary terminals by a factor as high as 1.73 (i have not seen greater) e.g. if your TF II-ry is 2 x 15 then your schematic may output up to (depends on diodes and transformer) 1.73 · 15V = 25.95V or when coils in series the double of that as 51.9V dc from 30V ac
... this however usually loads up to capacitor after many seconds when no load is connected
and even a moderate load such as 2kΩ brings it down near or even below of 30V dc
. . . so the only case in your case is that your battery forces your 317 to a low conduction mode that enables such voltage to build up at the input of 317
to verify that put a 1.2kΩ to 1.5 kΩ resistor shunting the input (wear protective goggles , place 1.5A fast blowing fuse before 317 e.c.)
-------------------------------------------------------
. . . other thing that now reminds is that the switched supplies may develop "waving" on output capacitors - that should be fading but the time it fades it may many times exceed the rating of the "filter cap." - thefastest way to overcome this is to put a current limiter before your filter cap
keeping in mind the function of your circuit then everything that is an overkill - i assume you intend to charge the "7.2Ah ups battery" their charging current is (must be) limited = see datasheet or a datasheet of a similar capacity , technolgy , peak load , battery-full voltage battery that actually lists the allowed charging current . . . . . . . . . and setup different circuits for initial constant current and later constant voltage charging - it goes complex but likely avoids exploding components (easier would be to buy any charger designed for your specific target - it's because what was just mentioned - you actually'd need to match your end voltage to battery temperature know the actual safe and effective currents and voltages that do cahrge your battery correct and keep it healthy)