Yes, as long as photo resolution is not (much) higher than video resolution.
Changing LiveView resolution on the fly might be possible; I had some half-successful attempts at that; however, capturing a full-res image takes time, so the video stream would be interrupted (you will lose a few frames, depending on how fast the resolution can be switched; rolling shutter time for a full-res image is about 130ms on 5D3).
One way to approximate full-res capture without interrupting the video stream would be to shift the "phase" in the 3x3 pixel binning (the initial row and column of each 3x3 pixel group). That
might be doable from ADTG and/or CMOS registers. If you have the patience to adjust these in adtg_gui and notice sub-pixel image movements, or maybe write a script to automate that process, you may find that register (assuming it exists). If you can also find a way disable the column binning, i.e. read out one single pixel out of 3 horizontal ones, on cameras other than 5D3 you would need 9 video frames to "assemble" one full-res image. Of course, that would only work for static scenes.
Without a way to disable the line/column binning, even if you can shift the initial line/column, recovering a full-res image is going to be difficult. Some deconvolution or super-resolution techniques might help, in particular if the scene is not exactly static when you take the picture.
Difficulty: give me
5 years and a team of researchers 