Like the SPORTIdent control units the recommended approach is to return the device to SPORTIdent or the local dealer.
However, like the control units it is possible to replace the battery yourself if you have some basic soldering skills.
The tip of the stick is sacrificed during the replacement so you will need a replacement tip in addition to a replacement battery (Renata CR1632FH1-LF). The replacement tips can be 3-D printed. Here is a design for the tip, you should be able to print (or get printed) from the stl file. If you still want to see the flashing lights then the tip needs to be printed in something translucent.
Phillip Herries from o-lynx.com has made this great instructional video about the process (he also created the design above).