You don't need the batteries as long as you leave them plugged into the USB-C the whole time. Since you need one side plugged in the whole time, it doesn't really matter which side you install the one battery... which side would be least intrusive to have a wire running on your desk? The main side (usually defaults to the left side, but can be configured to whichever side you want) does consume more power, so the battery won't last as long on that side.
If you'd like to use the Bluetooth to connect to multiple devices, maybe put the battery on the main side. The USB cable on the peripheral side is only for charging and flashing and can't be used for connecting the keyboard to be used as a wired keyboard. By default, plugging in the main side to a computer will switch it to a wired connection to just that device, though, again, whether the wired connection or bluetooth connection should be prioritized can be configured.
Here is a battery guide for the ferris sweep. : https://docs.beekeeb.com/other-guides/installing-a-battery-to-the-ferris-sweep-v2
The battery was the last part that I soldered on my corne build, though if you have a case you'll have to unscrew the case to do more soldering later.
be used to socket the board? And if yes: Do I solder the ones with pins only on one side to the pcb and the other ones to stick into the holes and solder to the nice!nano?