Since posting this, I bought a Kona EV, a Jupiter 1500W inverter, a 1500W space heater, 2 AWG jumper cables with alligators on one end and ring terminals on the other, and a 150 amp ANL fuse (which I wired into the positive jumper cable.)
With the Kona in Utility mode, the 12V system idles around 14.8V.
I connected the jumper cables to the battery's positive terminal, and to a bolt on the frame. The space heater ran for several minutes, as the battery dropped from 12.8V to 12.5V, at which point my ANL fuse blew. The Kill-a-watt was reading 1600-1700W at the time, but I'm not sure if it's very accurate with a "modified sine" waveform.
I plan to buy more fuses and test with a smaller load, but so far I can conclude that 1500W is too much. I want to find the power level that stabilizes above 13.0V or so.
I tried again with a hair dryer and two 200W light bulbs; the Kill-a-watt reads 1322W on the inverter (versus 1405W on grid power). I ran this on Utility Mode for 10 minutes, and the car's voltage only dropped from 14.66V to 14.64V. This is well above the charging voltage, so it could probably run indefinitely at that level. The infotainment display showed "Electronics" consuming 1.7-2.2 kW.
When I add one 100W light bulb, the car plummets to 12.9V and the Kill-a-watt reads 1535W, a puzzling gain of more than 200W. I'm not sure whether the last bulb pushed the inverter into a weird mode, or if the inverter was confused by the Kona's voltage drop, but the best I can say is: 1300W (plus inverter losses) seems okay, but much beyond that risks overpowering the LDC. If I were doing a permanent installation, I would probably use a 1000W true sine inverter and a 150A fuse.
I also read some docs on hyundaitechinfo.com. It describes the 2-wire "Intelligent Battery sensor" near the negative terminal, which tells the car how quickly the battery is charging/discharging. "When recharging the battery, ground the negative terminal of the booster battery to the vehicle body." I think the same logic applies to an inverter: you want to connect it directly to the body, and not to the battery's negative terminal, as that would bypass the sensor.