The video was extremely interesting

Great cucumbers too, but the lettuce and carrots looked as if they were struggling for light.
I use grow lights over a propagator to bring seedlings on in spring, but they never really get enough light. The growlights themselves are expensive but you could just buy the bulbs and make the reflectors yourself.
After poor light levels I think the biggest problem you would face would be temperature in the depths of winter, unless you have a heat generating trick up your sleeve.
My preferred approach would be to grow winter crops such as brassicas, lambs lettuce, japanese greens, sprouted seeds and so on for fresh winter food, and to rely on produce stored from the summer, either frozen, dried or stored in sand, to make up the bulk of your home-grown winter food.