Trackpad and keyboard not working after installing battery on 2017 Mac

Mmm… Two unrelated issues!

The internal keyboard and trackpad connect to the logic board independently, both connections are near each other. So you’ll need to inspect the connections for damage or the cables will need replacement. Reference this marked up image, I’ve marked the two connections

As far as your Bluetooth mouse, that’s clearly a mouse issue. Double check to make sure it’s being recognized within the Bluetooth settings, it’s also possible the mouse is having a sensor issue if it is responding at all.