I have been doing this with many different UPS units (including some just like the one you have). I wrote this article a while back and have modified dozens of UPS units to run on external batteries.
I have several 1500 VA units that are similar to the ones you have. They too have fans in them (unlike the 1000 VA models). In addition, they have a port at the top rear of the unit to plug in an external battery. This is essentially the SAME THING you are talking about doing; adding more capacity than just the internal batteries. I use the external port on mine to attach more capacity.
While these units are great, I have mostly stuck with using the older ones (which I get free at work when they throw them out) which run on 12 volts instead of 24. I mod them and then give them to local Ham Radio people to keep their emergency batteries for their radios charged up. And, bonus for them, they work as regular UPS units as well.