It's just expectable that a 600 VA UPS isn't effective in supplying a router with 5W. I guess it can be explained by looking at the technical data, the latest by measuring the no-load current consumption.
These PC UPS are designed to deliver relative large power for a short amount of time, e.g. 5 or 10 minutes, efficiency with small or effectively no load isn't a design goal, and not easily to achieve.
Choosing the right UPS is probably not well supported by the bold and simple advertising statements, it needs quite a bit thinking of yourself.