Lower end APC's are notorious for putting out a really chunky square wave.
Try stringing APC's like the SmartUPS 1000 in series, by the time you get to the third one, the output 'power' is useless, as it's been mangled so badly.
Try running a small electric motor off an APC 1000, you can here it chunking away, hating the wave form.
As you say, your upstream inverter is a nice true sine wave. That's what you want your gear running on.
It could be that your onboard NICs are behaving very differently to your PCI nics with respect to bad power. Different rails on the power supply perhaps.
The other distinct possibility is earth potential differences while on UPS. Some floating earth difference is drifting across some of your ethernet cables, and smashing your packets. Just a tiny leak or float on 230v is a big deal to 5v ethernet. Shielded ethernet can make the problem worse, better off with UTP unshielded-twisted-pair.
Make sure all your gear is earthed properly.
I think an oscilloscope is going to tell you a lot more than wireshark.