Sorry, but I'm confused. The onboard 4000 watt generator powers everything but the 5500 does not? That doesn't compute, so the difference must be something other than wattage.
The better quality portables are Honda & Yamaha, but the less expensive competitors such as Champion have improved to the point where they are worthy of consideration. I would suggest an inverter-type generator.
https://www.championpowerequipment.com/product/100519-6250-watt-open-frame-inverter/