You'd need the capacitor's value and ESR (and ESL and strays and other attached bypass, as applicable), and the diode's surge rating for time scales on the order of ESR*C.
The surge is not hard to calculate or estimate by hand, but SPICE can be used if you like. Helpful for complicated PDN (power distribution network)s.
The diode should also have Vf less than 1-2V under the surge condition, I suppose. That likely rules out 1N4148. May also rule out a lot of fast-recovery diodes, which tend to have higher Vf.
Ideally, you would check against the short-duration surge ratings for the diode. These are rarely given, let alone at the arbitrary duration and waveform a particular circuit would give.
In the absence of such ratings, it's not obvious whether a diode will survive such abuse (though it likely will).
The Vf constraint is probably a good start, and current, surge, energy or power ratings may not necessarily be exceeded if that one is met. Point is, there are multiple dimensions to constrain here, which cover slightly different directions, and which may have some overlap, but not total. YMMV.
It is probably okay to assume a diode of comparable size (i.e., a 1A diode like 1N4001 to complement a ~1A regulator) is adequate.
You could make a power supply shorting jig and test reliability, I suppose. If it passes 1k or 10k or however many cycles, at high and low temperatures, with the maximum expected load capacitance, it's probably fine.
(Note that a failure condition includes the diode or regulator going short or open. Who knows!)
Tim