Battleships were generally designed to resist damage from guns equivalent to the ones they were armed with: hence the importance of "immune zones" in design (the zone within which the ship was too far away for specified guns to penetrate the side armour, but too close for them to penetrate the deck armour).
The importance of rate of fire depends on the engagement range. Battleships didn't carry many shells (typically 100 rpg - not necessarily all AP) and they waited to observe the fall of shot so they could correct the aim before firing again; and at long range, shells could take well over a minute to arrive.
To sum up, battleship designers and navies were very well aware at the time of the importance of displacement: the need to achieve reasonable speed together with good armour protection and guns powerful enough to penetrate that armour kept pushing up displacements. Just look at the wild designs German naval architects came up with for "super-Bismarcks" with guns of up to 20" calibre. Why would they bother, if the Bismarck was so good? With battleships, bigger was better (other things being more or less equal).
This site probably makes the most detailed attempt at analysing the strengths and weaknesses of WW2 battleships, and it rates the Yamato as clearly superior to the Bismarck:
Battleship Comparison