I got a Tripp Lite UPS for my Lenny box, and downloaded the rpm of the power management software and converted it to a deb, but I haven't installed it yet. When I plugged in the USB cable, Lenny displayed the battery level of the UPS. However, it says it's running on battery power and discharging, even though it's plugged in.

I really don't want to test to see whether Lenny will shut down gracefully when the battery discharges, but I'd rather not install the software if I don't need it. Does anybody have experience with Lenny and Tripp Lite using the built-in power management? And is there some great feature in the Tripp Lite software that makes it worth installing anyway?

Then something isn't configured properly.
If you don't test it, then you won't know if it works or not, until it is too late

The UPS is there to keep your PC running for long enough to shutdown gracefully in the event of a power failure. When properly configured it could warn your users "There's a power failure, this system will shut down in 3 minutes. Please save your work and log off NOW".

Make a backup. Test what happens when the UPS battery fails.
Otherwise there's absolutely no point in having it and you have wasted your money.