Is UPS wattage rating for battery backup outlets only

You appear to not be sure that one or more of the devices connected to the ES700 is the cause of the issue. So, eliminate them all as a potential suspect... or reveal them to be the trouble.

From your setup, you have a heavier load and demand on the lighter UPS. The ES550 is a 330 watt UPS, and the ES700 is a 405 watt unit. The PC, TV and Monitor should be drawing far more power than the game consoles and network switch. So, take a night for this little experiment. Plug all of the consoles and the network hub into the ES550. See if they cause the overload alarm to trip in that unit.

Now... to be sure, you would need to do the reverse as well. That means plugging the PC, TV, and monitor into the ES700. If the unit is fine, there is no reason why this should not work. If the unit is not fine... as in the unit (not the battery) is now defective or broken, this could put your devices at risk... so make sure that your APC warranty is good and will cover any damage to your devices.

I also understand this might not be possible to do... this kind of test, since the ES550 has different output plugs from the ES700. That is unfortunate, since from what you are describing, the problem might not be the consoles. The problem might be the DC power adapters that the consoles use. Even if the consoles are not on, the DC power adapters are still drawing power.

As David mentioned, the battery is almost certainly dead if this is an older UPS, and this is confirmed by an 11v measurement when you know the battery should be fully charged ( when it should be 13-13.8v ). More importantly, a 200 VA UPS is so weak as to be useless. Modern PCs have 350-500 watt power supplies ( more for extreme rigs ), so you need at least a 350-500 VA UPS, especially if you want to also have the monitor plugged in ( to a battery backed outlet ). I have had a 1000 VA unit ( extra capacity -> more runtime ) for 14 years and have had to replace the dual batteries every 2-3 years.