These types of lights with the 9V to 30V range have a buck current regulating circuit inside, so there should be no issue hooking it up to your jump starter, except that jump starters are typically not very high Ah capacity so you will have a limited amount of time to run them from it for testing.
Your total power draw is marginally excessive unless you upgrade your alternator. 540W total / 12V = 45A. The actual total may be a bit lower. Lights like those are often fraudulently rated based on # of LEDs * 3W per LED to arrive at a wattage claim, while the actual drive current is reduced to ease heatsinking so they can make the enclosure at lower expense. This is more clear with the "spot beam/bars" which if they have 48 x 5050 LEDs (which are 300mW each max rating), that is 14.4W, not 120W. Perhaps they are using shady ratings and "forgot" to elaborate that the 120W figure was really an incandescent wattage equivalent. In fact, 14.4W could easily be very near a 120W equivalent using existing LED tech.
This factor makes a proper calculation quite difficult. You could use an ammeter to measure the current of each light from your vehicle electrical system, "temporarily" connected directly to the battery, though even then it would be nice to have fuse in case something goes wrong.
Once you have an amp reading, you can calculate total current draw and determine the rating and number of automotive relays to use. You'd ran the battery to the relay contact, the other relay contact to the light bar(s), and use a second run from the battery, to relay coil, then the other relay coil terminal to the switch in the cabin, then the switch to any chassis ground point. The negative from the light bars would go back to the battery negative if they're remotely as high current as specified rather than a chassis ground.
Once you determine the total current, then you can pick an appropriate wire gauge.
Personally I would be going over all the light specs again because there's no way that 48 5050 LEDs are going to produce 12K lumens, and if the other light has false specifications as well, you may have overpaid or at least are supporting fraud. If this does not bother you then you still should determine the true power consumption for the purpose of not overspending for more relays and wire gauge than needed to get the job done plus a conservative margin above that.
However it is not much more expensive to get a higher current relay so you might just get 2 x 30A relays and run the so-called 300W light bar from one and the pair of spot beams from the other. The two relays would simply be wired in parallel except of course for each output going to the respective lights they're powering, and being automotive relays you'd expect and should make sure they have 12V rated coils.
Previously I suggested powering these from your vehicle electrical system because it, with engine running, should have a more stable voltage than whatever the jump starter is producing, and then you'll have roughly 14V to the lights but should take the current measurement and then factor for 12V if you wish to run these without the engine running, except that this is quite a power draw to have for a typical starter battery without the engine running so it would be best not to use them with the engine off.
As far as what your existing alternator is capable of, "IF" they do truly draw 45A, your alternator will not be able to keep up unless your engine RPM is above some threshold higher than idle speed. There are too many variables for me to determine the exact speed and it may not be worth bothering if the lights aren't even rated accurately and end up drawing a lot less current than the specs suggest.
You might check with the seller to see whether they also sell a wiring kit. It may be made of garbage quality components but at least you could ask what is in the kit as far as wire gauge or relay ratings. If they can save a dollar building the kit for the true need rather than the questionable ratings they might do so and so you would see what they're thinking is enough current capacity.
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