I'm having a little trouble teasing out exactly what your question is, so I'll just write some things about sheaves that seem related and hope they are helpful.
Suppose $C$ is a site. Let $\hat{C}$ be its category of presheaves and $\tilde{C}$ its category of sheaves. The topology defined in SGA 4, II.5 is on $\hat{C}$, not on $\tilde{C}$ as you suggest in your question. Its purpose is to give a topology on $\hat{C}$ such that the category of sheaves on $\hat{C}$ (that is, contravariant functors satisfying descent on the category of contravariant functors on $C$) should coincide with the category of sheaves on $C$ (i.e., $\tilde{\hat{C}} = \tilde{C}$.tilde{C}$).
You've got the condition for being bicovering backwards: a map of presheaves $H \rightarrow G$ is called bicovering if it is covering (with respect to the topology on $C$) and its diagonal $H \rightarrow H \times_G H$ is also covering. (What it means for a map of sheaves to be covering is that for any map $X \rightarrow G$ with $X$ representable, the sieve of $X$ induced by $H \times_G X$ should be covering.)
A Grothendieck topology on $C$ is described by asking certain subfunctors (sieves) of objects of $C$ to be covering. If $H$ is a subfunctor of $G$ then the relative diagonal map is automatically an epimorphism since it is an isomorphism (by definition). The covering sieves of $X$ are the subfunctors of $X$ that become isomorphic to $X$ upon passing to associated sheaves.
The bicovering business arises when one wants to study which arbitrary morphisms of presheaves (not just inclusions) become isomorphisms upon passing to associated sheaves. The notion of a covering morphism of presheaves explains which morphisms become surjections of sheaves. The question then remains: which morphisms become injections? A map of sheaves is an injection if and only if its relative diagonal is a surjection, so the condition is that the relative diagonal be a covering map.
