Personally, and since I've met this effect while simulating "power systems" *, I would just add input capacitance, also making sure the resonance frequency is far away from any "power" oscillations (such as PWM signals) used in the system.
* Ideally we want caps with as low ESR (and ESL) as possible here. Caps are charged/discharged in a controlled way.
Depends on what "power systems" include, but this can be undesirable as well. Startup and turn-on transients can very easily excite unintended resonances, so it's always good design to make sure all resonances are well damped. If nothing else, you can't always guarantee that the only inputs to, or outputs from, the system will be fully under control of the control circuitry. E.g., say you slew-rate limit the circuit, so it can't experience startup transients under normal conditions. But someone can always short-circuit the output, forcing a transient there. This can extend to, yes, rather unlikely stimuli (e.g., a diode fails shorted, discharging the filter network?), but the "best practice" conclusion is that, such situations should be avoided, by design, in the first place.

For an analogy, it's the difference between a car that drives absolutely smoothly, versus one that has some stupid rattle or vibration when the engine is at a particular RPM and load. Or suspension that rides over bumps just fine, but sways just a little nauseatingly (a common mode - differential mode problem).
What I like to do with DC link filters is, put on enough electrolytic bulk as needed for filtering / ride-through, and enough film capacitors to handle the high frequency switching ripple and to get a nice low loop impedance going into the inverter (which may have additional inductance added intentionally). The electrolytics don't have vanishingly small ESR, so they act to dampen the film caps (which, depending on how they are connected by traces or pours, may act essentially like one solid bulk capacitor, or may exhibit some lumped-equivalent transmission line behavior; in either case, ESR plays a role in damping it).
You generally want to avoid multiple levels of bypasses, because you don't need it, and you don't want it. Placing small ceramics right at the inverter results in an additional LC loop, which probably isn't well damped because sqrt(L/C) will be higher than the surrounding impedances, and a suitable amount of series inductance
is desirable in the inverter.
Tim