It’s not hard to check that the subobject classifier $\Omega$ is trivially fibrant, and so its homotopy type is trivial. Orthogonality of $\Omega \to 1$ against a map $i : A \to B$ corresponds to the property that any subobject $A' \to A$ extends to a subobject $B' \to B$ such that $i^*B' = A'$; when $i$ is mono, this is always possible, with canonical solutions given by the direct image $\exists_i A'$ and the dual image $\forall_i A'$.
Nothing here is special to simplicial sets: this argument anpplies equally in any model structure on a topos in which all cofibrations are monomorphisms.
This argument appears (tersely!) in the proof of Theorem 1.4.3 of Cisinski 2006, Les préfaisceaux comme modèles des types d’homotopie (Presheaves as models for homotopy types), 2006 (thanks to Tim Campion in comments for the precise reference). A couple of closely related arguments — firstly the fibrancy of a universe classifying certain fibrations, corresponding to showing that those fibrations extend along trivial cofibrations, and secondly the univalence of this universe — appear in my 2012 paper with Chris Kapulkin, The simplicial model of univalent foundations (after Voevodsky), in Sections 2.1, 2.2, and 3.2.