Wow, so much stuff here, most of it seemingly good. Let's not forget the OP's original problem, that the BB had some free play. Of course you need the correct spindle or one not so very different, but solving the free play problem requires knowing where it coming from. A spindle of the wrong length won't give it free play. A mismatched taper might if the crank arm bottoms out against the flare at the inner end of the taper. Something as simple as poorly adjusted bearings certainly would. Wrong cups or BB width or bearing size could also if they prevent good adjustment.

Start with the simple stuff. You already have a crank and spindle installed. The chain line may or may not be okay. Start by holding one crank arm while you try to wobble the other. If the whole assembly, i.e. both arms, moves then it's in the BB bearings. So try to readjust the bearing cups. If one arm can be made to wobble even when the other is held firm then it's in the arm/spindle joint. Try it on both sides. If you really need to replace the spindle then you need to find the correct length and symmetry and taper, etc. It isn't rocket science.