I have no specific experience with this cold brew device but a general understanding of how extraction works and experience with other devices. In short, you can definitely use water at room temperature to make cold brew (I think it is better), but there seem to be some specific issues with this cold brew maker.
In general the colder the water, the slower it will extract the solid compounds from the coffee. This is the main thing you have to keep in mind, because you will have to adjust some other variables in order to reach the right extraction. In their instructions they suggest you use a medium grind level. This is one variable you could change. Instead of a medium grind level I would use a medium to coarse grind. Another variable that you could change is the brew time. They set the device to 1 drop per second but you could increase that a little to speed up the brewing process, resulting in less extraction.
Specifically with this device there seems to be the problem, that air bubbles block the valve (see the FAQ on the website). They argue that the changing water temperature is responsible for the bubbles to appear. However if you brew at room temperature the water temperature should be much more consistent than with ice water which will obviously melt eventually and then the water will warm up. So I'm not quite sure about their logic (unless you are supposed to add ice until the brewing is over).
I suggest you go ahead and try it at room temperature, adjusting the other variables. Make sure you pre infuse properly so the grounds degas as much as possible, that should prevent too much gas blocking the valve. Alternatively you could use refrigerated water and brew it in the fridge.
Hope this helps, in the end it's a lot of trial and error anyways. Have fun brewing.