I think you can find an answer in my question about global fibrations of simplicial sheaves: global fibrations of simplicial sheaves .
There, Andreas Holmstrom pointed me to Voevodsky's preprint "Homotopy theory of simplicial presheaves in completely decomposable topologies", https://arxiv.org/abs/0805.4578 , where I discovered lemma 4.1 , which I think aswers your question.
Although it is not proved, Voevodsky says that it is straightforward. I could manage myself to write a proof of it, at least under the hypothesis of Brown-Gernstern's "Algebraic K-theory as generalized sheaf cohomology", in LNM 341/1973, that is, with sheaves defined on a Noetherian space of finite Krull dimension.
In this situation at least, fibrations are global fibrations. That is, a morphism of sheaves $p: E \longrightarrow B$ is a global fibration if and only if if for every inclusion of open sets $U\subset V$ the natural map $E(V) \longrightarrow B(V) \times_ {B(U)} E(U)$ is a (Kan) fibration of simplicial sets.
As a corollary, if you take $B = *$, this condition tells you that fibrant objects are those for which each restriction map $E(V) \longrightarrow E(U)$ is a Kan fibration. In particular, put $U=\emptyset$ and this implies that each $E(V)$ must be a Kan complex.
All this with Brown-Gersten's hypothesis, but Voevodsky doesn't seem to need them, so maybe it is also true in your situation.
