Can't help you with the Park tool, but the clicking sound could be a pit in one of the cups or if it is a bit of a creaky click, perhaps wear on a crank. With the chainset off, you can feel the state of the bearings quite well just by turning the axle in your fingers. If that is smooth and well adjusted, the problem could be with the axle/crank fit. 2-3mm movement suggest all is not right within the BB. When it is all out, check the axle bearing surfaces carefully too; these can also become worn beyond adjustment and it's so much better to know before you've reassembled the bearing.