Unfortunately, to fix the problem you need to unscrew the small screw which holds the key halves together. The threads are coated with blue thread locking fluid. If that's Loctite, it's supposed to be possible to disassemble using normal tools. Unfortunately it's not very easy. You need a good screwdriver that fits well, a lot of downward force to prevent slipping, and strong turning force. Heating the screw with a soldering iron might help, and if done carefully, it doesn't seem to damage the plastic. I first got the screw to turn a bit, but it wouldn't go any further. What finally helped was spinning the screw back and forth a bit continuously while unscrewing.
This is a ridiculous design. Changing a battery shouldn't be that difficult. This deserves a recall.
If you want more information on the procedure, here's another guide. The only other tricky thing I encountered was separating the two key halves at the key ring end. They refused to simply pop apart, but wiggling from side to side (ie. rotation around the axis through the key ring hole) allowed the halves to separate. The battery is a CR1616. Some parts of the photo are blacked out because I don't know if any of those numbers are codes for remote unlock or the immobilizer.