That reader looks very similar to the one that used to be sold here on DT (but isn’t necessarily identical, they all have the same case but sometimes different guts). You can use that one, but you have to be very careful about getting good coupling while you write to your tag. The antennas in those are not shaped well to couple with the cylindrical coil in the xEM & NExT, and writing requires a much better coupling than reading. Poor coupling during a write can cause tearing in the data and brick the LF side of your tag. If you do decide to use that cloner, make sure to take the case off and test with a LF xLED exactly where to place the bare antenna to get the best coupling, before you attempt the write to your NExT.
Regarding the HID reader at your work. Getting the orientation just right for the small cylindrical coils in the NExT and xLED tester can be tough. If you had a diagnostic card, you could see it light up when you place it near the reader. You can try with the xLED, but you need to get the orientation just right. Check out this post.
Even if you do get the orientation down, I’m not sure you’ll get any reaction from the reader when you scan an unregistered NExT tag. Many readers I’ve encountered will read the tag when you present it, but only flash the lights or beep if they recognize the tag from the directory of whitelisted/blacklisted UIDs.