Alternatively, in broad terms what I should do to achieve this?
I'm also new to Bullet, so these will be very broad terms:
I think you may want a kinematic object. Kinematic objects are controlled by the application and do not move when other objects collide with them, as though they have infinite mass.
However, that also means they might not be stopped by static objects such as walls, so you might want to create a constraint of some kind, maybe a very stiff spring, that is attached to the stick and to the world, and the user can move its world attachment point, and it will drag the stick along with it. With the stick as a dynamic rigid body, it will be stopped by walls or other objects, instead of shoving them out of the way like a freight train.