On this point, some batterypacks do have both dc and ac inverter output as well, same as cars that have as inverter. But as far as useful for cooking they (or more specifically the underlying battery) are typically limited to 10amps which is 120watts total power and after inversion losses is more like 75 to 100w usable watts. The intention is this is for small devices such as charging laptops. This limits you more to easybake oven cooking vs imagining you had a microwave/hairdryer.
Bootwarmers even fan-driven powered bootwarmers that run at 30w-45w, need like 30min+ to get room temp boots (70F) up to body temp (100F); In a ice cold car, this 30w is not enough to keep boots warm, you need to keep the boots insulated in a bag and even stuff your gloves in the top otherwise they lose heat faster than the 30w the heaters are putting in..
On the point for hotlunch, The ac inversion would be very inefficient and typically 30%+ of the battery is lost in the process to heat. Using a powerpack for dc it will also still lose maybe 10%+ to heat too. But if the idea is you are using this electricity power solely to maintain temp anyway, heat is not a total loss, so just throw the battery pack into your warm bag too.
This is the same concept as on the slopes with cellphones that dying in the cold to use a batterypack charger; the charging process also provides heat as a byproduct in both the batterypack and the phone.
Do keep in mind these are all lithium batteries have a risk of blowing up or spontaneously catching on fire, especially when left unattended in an enclosed hot environment, at higher drain settings; and if it is some random thing you found on amazon.
I'd still suggest your biggest loss is via the container and insulating and keeping this hot thing hot; so using a purpose built insulated small cooler/lunchbag to hold heat IN is where you will get the most bang for your buck. (You can test by putting this lunchcooler contraption into your freezer or refrigerator). To round out the above parallel example, this is also the same as if your iPhone is in a phonecase, it also won't freeze up as fast, and holds the heat from the charger trick longer than a naked iphone.
The heated ski gear bag I do not believe is focused on insulation vs its secondary use as drying things and letting the steamy air out.