When an object floats, its weight is balanced by the buoyant force. The buoyant force is equal to the weight of liquid displaced by the object. From these two arguments, we can say that the weight of liquid displaced is equal to the weight of the object. This means that, it makes no difference to the hydrostatic pressure in the system upon replacing a certain volume of liquid by an object of same weight of the replaced liquid.
...because of the buoyant force exerted on the mass by the fluid, there must also be an equal and opposite force exerted on the fluid by the cube.
Let me make it clear, that introduction of the floating object doesn't make any pressure difference. You're right that there must be an equal and opposite reaction force to the buoyant force as per Newton's third law. However, even when there was no object, the liquid below a particular volume exerts upward force equal to the weight of liquid above it and has it's own action-reaction force pair.
But if we take the increase in the level of liquid in the container (due to the liquid displaced), then the pressure increases. The effect of placing an object is same as pouring an equal weight of liquid into the container. The following diagram illustrates this fact:
The hydrostatic pressure in all the four cases in the following diagram are equivalent:
In short, in fluid statics, the pressure depends only on the level of liquid in the container and is independent the object floating.
Image source: My own work :)