There should be a difference in power consumption between a) not performing any graphics related tasks and b) putting the GPU, the video memory and the HDMI port all into suspend mode - similar to sending a smartphone into suspend mode, while keeping the CPU running.
How much power saving b) will amount to, I don't know. Will it be closer to 5%? Or closer to 30%?
The portable power bank you linked to above is a little big for my use case. I would prefer to use a smartphone battery to power, say, a single core device. Do you think this would be impossible?