How can I download the timelapse videos while the camera is in privacy mode? I don’t want to have the camera on and recording just to do that.
Welcome to the Wyze User Community Forum @Obs! 
With the card mounted in the cam, you can’t. Privacy Mode for the Panv3 = OFF. No Wyze Cam can process a Timelapse video download while it is off.
However, if you pull the SD Card and read it in a computer or external reader, you will find the Timelapse video file within the Timelapse folder in a .h264 file format that will need to be converted to .mp4
What’s the best way to get at the SD card in the camera? When it’s off, that side of the camera is against the wall, and I’ll have to unscrew it on top. Is there a mode to put it in to get at it?
If it is mounted such that it obscures the SD slot when it is in Privacy Mode, you should be able to manually manipulate the pan & tilt to bring the SD slot to the front and pull the card while it is on. It won’t hurt the card or the files to pull it or insert it while it is powered.