I found a couple of Duracell 303/357 batteries in my drawer (presumably they are suitable replacements for SR44). They worked!
There's a curious vibrating noise after the shutter release. I hear the noise after both the opening and closing slaps of the mirror when holding the camera normally, but when I hold the camera upside down or sideways, the vibration disappears on the closing slap. The dampening foam strip where the mirror slaps against the top appears to be in good shape, so I suspect the vibration is caused by the flat part (sensor?) that hangs behind the mirror (attached to the mirror hinge). Because it moves in tandem with the mirror, but has some play, I think it vibrates briefly against the mirror. Is my hypothesis correct? What's the name of that part? Is it normal for it to vibrate briefly after a mirror slap?