I don't remember all the technical details, but basically a simulated sine wave apparently can put a lot of stress on a PSU; I don't recall if it is specifically IN the PFC circuit, but I think it may be BECAUSE of the PFC circuit. The PSU thinks the input power is so bad that it shuts off.
ANYway... I have an Antec SG-650, I still believe one of the best PSUs out there, even though its efficiency has been passed by. It will not run on a typical UPS (e.g. APC Back-UPS), shutting down immediately, just like yours. I got a true sine wave unit, also APC (Smart-UPS), and the Signature is entirely happy with it.
I have not tried any of the new Cyberpower PFC models, but I suspect that would work, IF its waveform is indeed close enough to true sine. If you get one, I'd be VERY interested to know how it works out for you. Even if this thread has died of old age by then, please PM me about it. Thanks.