Is it possible to take take a picture while recording a video?

If you think of one single shot within the video, it should be doable by only setting the time stamp of your request. The picture, on the other hand, would only be available in the (rather poor) quality of your video. Requesting a picture of higher camera quality from within the video stream would in any case affect quality of the actually recorded video.

Pity. I was hoping for something along the lines of, an image stream comes from the camera in the camera's native resolution and gets downscaled before encoding into a video stream. If we vedge a hook between the camera and the donscaling and encoding process, we might be able to get a full resolution picture. Form what you say, my hope was too naiive. Pity.

So I guess the only way is to interrupt the video recording, reset the camera, take a picture, then start video recording again, right?

The reason for downscaling resolution of the single pictures of a video is only the sample rate, as gathering data from the sensor takes its time. The higher the sample rate the lower the possible resolution, simple as that. That's why the video's resolution is always lower than the picture's when using the same camera.

Yes but does it come already downscaled from the hardware or is it downscaled at some later stage? If the latter then a hook should be possible. Saving the picture could be done at a lower priority, it does not need to be in real time.

As I am lacking the necessary insight into development of cam sw I can only guess. I think it is already sampled in a lower resolution per frame. It would not make sense to me the other way, thinking of possible performance with this phone. And also in this case you would need an additional step in the gathering routine to save the single frame of desire, which would propably slow down the main process, again resulting in quality loss.
But congrats, you really got my mind spinning ;)