How do you wire something that opens
Normal multistrand cable tolerates well bending and torsion if the bending radius is large enough and the torsion isn't too many degrees per inch.
As suggested have hinges or other mechanical limiters which keep one edge of the roof nearly in place when the roof is opened. In that side have some binding that keeps the power cable on the surface of the wall.
More preferably you can let the cable run out of the box or behind a doubled wall. Then surely it's not in the way of the radiation and can have so much extra length that the bending stress is neglible. Having no radiation stress on the cable lets you select the mechanically best cable.

Hopefully the next image with some transparent walls can show how a cable could run freely inside a doubled wall (red) (only the problematic lamp set is drawn):

The cable will not be bent with tight bending radius, the bending distributes along the the length in the gap.
As the third principle you can put the cable have a long run near the hinge. The  fasteners must be loose or preferably the run is in a tube. The motion is mostly torsional and not especially many degrees per inch.

If you are going to make a device for someone else you meet a problem. The cable must be mechanically protected and the roof movements mechanically limited so that no special carefulness is needed. The first version must have an armor around the cable. It cannot be recommended for other use than as your own custom device. The 2nd and 3rd versions are less sensitive.