Among non-competition automotive applications, the place where high power output relative to battery size (or energy capacity) is needed is in hybrids: they need to put out enough power to drive the vehicle, despite being small.
When the energy capacity required is really small (a classic hybrid that can only move a few blocks on the battery alone) nickel metal-hydride (NiMH) is still viable - Toyota still uses it quite successfully.
When more energy is required, for plug-in hybrids (and the logic would apply to short-range EVs), lithium-ion is used, but in specific electrode chemistries for power density (at the expense of energy density) and with sufficient cooling that it survives. That is a reason for the popularity of Chevrolet Volt batteries in relatively high-performance DIY conversions: in addition to being plentiful, and having been around for a while, a 16 kWh Volt battery can put out 120 kW for at least 10 seconds at a time even by GM's conservative ratings. The LG Chem modules built for the Chrysler Pacifica have been popular as well: plug-in hybrid power capability, and while the Pacifica Hybrid is rare compared to the Volt, the modules (new) were sold by various suppliers. There are other plug-in hybrids, although none are as common as the Volt and none of their modules are as available as the Pacifica modules were.
When the supply of Pacifica modules was cut off, EV West apparently went looking for something else they can buy and found the JH3 modules. They have apparently found them suitable at high C-rates; my concern is that this is not endorsed by LG Chem and this isn't even the ideal choice from among the ESS product line for high discharge rate. Maybe they'll work at repeated sustained high discharge rate, but I'll bet EV West won't warranty them for that.
The other solution is simply to use a big battery. Any cell technology is limited in its discharge rate relative to its capacity, so if you double the battery size you get double the power capability, even though you don't need the energy and don't want the weight or bulk. It's not a coincidence that when two Tesla models have the same motors, if one has higher rated power it also has a bigger battery. You might hope for a Model 3 Performance with the less expensive and lighter standard-size battery, but they know that won't work so they won't sell it to you (and they have more mercenary reasons as well, of course

).
There have also been companies offering high-power cells for drag racing applications. Maybe they are still operating; they're certainly not relevant to reasonable street-driven EVs.