Crank sliding 1 mm right left
There's been a problem in your situation with preloading the bearings is my guess.  Based on the external bearing cup and a glimpse at the end of the video of your left crank arm, it appears you have a SRAM crankset installed? This video from SRAM, notes that insufficient preload of the bearings before full torque is reached may result in lateral play of the crankset.  This is shown around the 3:30 mark of the video.  While this video features a non-BBright  bottom bracket shell, the similar problem to yours is fairly meaningful and if not the exact reason or answer for your case, I think it'll be along those lines. 

The other video that's linked at the end of this paragraph features BBright bottom bracket and compatible FSA crankset install. Significant here is that with this particular crankset and others that mate to the BBright system, bearing seals and wave washers need to be placed and a fair amount of torque--around 45-50 N-m--is required for the fixing bolt.  Missing the washer placement or less than adequate torque will lead to symptoms such as yours.
BBright bottom bracket and FSA crank install

If a LBS did the BB install, it's best to return the bike to the shop and discuss it with them.  The rework should incur no additional cost (IMHO).