Much of the discussion seems to be about side to side play. And I don't know how one really tells that is going on when they pedal. Al least I don't think I'd notice a millimeter or two, or even three maybe four mm of side to side play in the crankshaft as I pedaled.

The GXP cranks have a smaller diameter shaft on the left side. Or at least some do. At first I was wondering if your BB wasn't for cranks with a 22mm diameter on the left side. And instead it was 24mm on both sides. Which would have the left side notably flopping around as you pedal.

The drawing clearly says the TK888 is for GXP cranks. And that it's for a 68mm shell. If your's is 73mm. Then I wonder if the play you are feeling is simply the crank arm not being fully on the spindle because there isn't enough of it sticking out the other side. And the crank arm is wobbling on the shaft as you pedal. Possibly destroying the splines on it.

Especially if you installed the 1mm spacers on either side as shown. Though even leaving those off will only be good enough for a 70 mm shell. Also, that BB has a adapter you are suppose to add to that left side to bring it down to 22mm diameter on the left. If that is a GXP crank that is different diameters left and right, then it needs to be part of your assembly.

You need to find out why they gave you a BB that only shows a installation drawing with a 68mm BB shell.