Surprisingly and unexpectedly the battery already arrived today! 
I placed it towards the left side and fixed it with a curled up cable tie. No grinding was necessary, but I removed a tiny little sticker inside the battery compartment (see below).
Below are some size comparison images and and the temperature measurements (they don’t work).
(This is the sticker I removed.)
No matter how hot the CPU and SOC are, the battery temperature doesn’t change. I therefore conclude that the temperature monitoring does not work.
The battery does its job well however, and it didn’t fail me in a 40 minutes phone talk that I just had. I will also try the charging at night and will report any weird behaviour.