Which capacitor type is best suited for pulsed power applications?
To answer this, you look in something called the "datasheet" for each of the candidate capacitors.  In particular, parameters like the maximum ripple current, maximum pulse current, and equivalent series resistance (ESR) will be most relevant.

You may not find all these specs in all datasheets.  Maximum pulse current is what you really want to know, but that is rarely specified.  Capacitors with low ESR will have a ESR rating and usually a ripple current rating.  The RMS of the ripple current times the ESR causes heating in the capacitor, which is usually what limits the ripple current spec.  Calculate your RMS ripple current and eliminate any caps for which it is too high.

The instantaneous pulse current may be limited by other effects.  Capacitors intended specifically for this kind of application will probably have a maximum pulse current spec, but otherwise probably not.  If you can't find this spec, then identify a few capacitors that have a suitable ripple current spec and otherwise seem appropriate, and contact the manufacturer.  Unless you find capacitors specifically intended for high pulse applications, this is a sufficiently off-mainstream application where you should have a conversation with some apps engineers from a few manufacturers.