Because in reality both animals implicitly know that the conflict policies are somewhat hardwired as: 'If you push me too far, I will fight you on the beaches.'

Knowing that that is the policy that is implemented, and that fights are always risky, whether a fight occurs will be determined by the reward size and the predictability of the outcome.

Yes, and both creatures deduce the predictability of the outcome primarily from the results of... a simple size comparison.

I would think most fights in the animal kingdom are predation, where the prey faces death and the predator, starvation. Perhaps you had just dominance fights within the same species in mind?

Predation is an example of a fight that is NOT avoided. I have all species in mind. I am almost directly quoting Dawkins’ “The Selfish Gene.” A raccoon does not attack a grizzly bear because of a simple size comparison. Thus, the lives of both animals are spared and a fight is avoided.