Both axes do matter to the extent that you want a neutral vertical or compressive load on the support stand with no moments in any other directions. Ideally any tilting would be done about the CG of the box.
That is correct. But usually the top hat location is a given (unless it is DIY-added to a cab) and is assumed to be inline (or as practically close as possible) with the COG.
So when considering tilting using the top hat, the only parameter that is 'left' to be checked is the COG height, as it determines the arc that the top hat will draw (about the COG) and thus the amount of the COG shift that will develop. That distance should be compensated by the tilter in order to keep the cab properly balanced on the pole.
This is exactly what the BT-12 does, but because it is true only for a specific range of COG heights (by design), the COG height of a cab should be checked to insure that it is within that range and so will be correctly balanced when tilted by the BT-12.