hi,
Really out of my depth here, I'm not a 'power' man, and I've never worked on such a device. Just so you know.
As a back up system, I would assume it is fitted between the public power grid, and the local grid (of a shop/house/factory/hospital etc..) and as you said, only operating when there is no external power from the public grid, ie: a power cut. I assume its a comercial device, so I can only assume that the actual 'inverter' part is ok. I find it hard to believe a company would sell such a device with no over-load protection, so it should be something simple, maybe a circuit breaker has blown? Or a relay?
How much power (on average) are you drawing from this/external power? Because, say you've got your 2 and 2.5kW inverters, the main grid is good, and you're drawing 6kW, then the public grid shuts off, your inverters cannot supply that much power.
Maybe its the connection to the external grid thats at fault. If it doesn't know when theres a power cut, it won't turn on. So the 'automatic' part might be at fault.
Have you tried shutting off the external mains while the inverters are in operation? With, say, only 500W power being drawn?
Sorry if this is of no use, but problems are there to be solved, and 'brainstorming' is what I do.
BuriedCode.