I will try to answer the question. As I said in a comment, the Thomason model structure on $Cat$ is not simplicial model structure. Let $C$ be a small category, we will view it as a topological category. Denote by $C[C^{-1}]$ the topological category where we invert all maps of $C$ such that $C\rightarrow C[C^{-1}]$ is a cofibration of topological categories, then the coherent nerve $N_{\bullet}C\rightarrow N_{\bullet}C[C^{-1}]$ induces a weak equivalence of simplicial sets. Notice that $C[C^{-1}]$ is an infinity groupoid.

Let $C$ be a cofibrant topological category. The mapping space $map(C,D)$ in the model category of topological categories is given by the (standard) nerve of the following $HOM(C,D)$ category :

$\underline{Objects}$ are topological functor $F:C^{op}\times D:\rightarrow Top$ such that for any $c\in C,$ $F(c)$ is equivalent to a representable functor $D(d,-)$ for some $d\in D$.

$\underline{Morphisms}$ in this category are natural transformation $H:F\rightarrow G$ such that $F(c,d)\rightarrow G(c,d)$ is a weak equivalence for all $c\in C$ and $d\in D$.

Let $S^{1}$ a simplicial model for a circle. Let $k: sSet\rightarrow sSet$ the cocontinues Joyal functor which take $\Delta^{n}$ to the nerve of the groupoid with $n+1$ objects and only one isomorphism between any two objects.

Recall that $\mathfrak{C}: sSet\rightarrow Cat_{\Delta}$ is the left quillen adjoint to the coherent $N_{\bullet}$ betwen the joyal model structre on $sSet$ and the Bergner model structure on $Cat_{\Delta}$

Now $k(S^{1})$ is a simplicial set, and the cofibrant topological category $|\mathfrak{C}[k(S^{1})]|$ is an infinity groupoid and its cohenrent nerve is equivalent to $S^{1}$.
The finial result is that $HOM(|\mathfrak{C}[k(S^{1})]|, C[C^{-1}])$ is a model for $\Lambda C$, since the nerve of $HOM(|\mathfrak{C}[k(S^{1})]|, C[C^{-1}])$ is equivalent to $ map(|\mathfrak{C}[k(S^{1})]|, C[C^{-1}])\sim Map(S^{1},N_{\bullet}C[C^{-1}])\sim Map(S^{1}, N_{\bullet} C)=\Lambda N_{\bullet}C$

N.B. The only point that I did not explained is the construction of $C[C^{-1}]$.

I will try to answer the question. As I said in a comment, the Thomason model structure on $Cat$ is not simplicial model structure. Let $C$ be a small category, we will view it as a topological category. Denote by $C[C^{-1}]$ the topological category where we invert all maps of $C$ such that $C\rightarrow C[C^{-1}]$ is a cofibration of topological categories, then the coherent nerve $N_{\bullet}C\rightarrow N_{\bullet}C[C^{-1}]$ induces a weak equivalence of simplicial sets. Notice that $C[C^{-1}]$ is an infinity groupoid.

Let $C$ be a cofibrant topological category. The mapping space $map(C,D)$ in the model category of topological categories is given by the (standard) nerve of the following $HOM(C,D)$ category :

$\underline{Objects}$ are topological functor $F:C^{op}\times D:\rightarrow Top$ such that for any $c\in C,$ $F(c)$ is equivalent to a representable functor $D(d,-)$ for some $d\in D$.

$\underline{Morphisms}$ in this category are natural transformation $H:F\rightarrow G$ such that $F(c,d)\rightarrow G(c,d)$ is a weak equivalence for all $c\in C$ and $d\in D$.

Let $S^{1}$ a simplicial model for a circle. Let $k: sSet\rightarrow sSet$ the cocontinues Joyal functor which take $\Delta^{n}$ to the nerve of the groupoid with $n+1$ objects and only one isomorphism between any two objects.

Recall that $\mathfrak{C}: sSet\rightarrow Cat_{\Delta}$ is the left quillen adjoint to the coherent $N_{\bullet}$ betwen the joyal model structre on $sSet$ and the Bergner model structure on $Cat_{\Delta}$

Now $k(S^{1})$ is a simplicial set, and the cofibrant topological category $|\mathfrak{C}[k(S^{1})]|$ is an infinity groupoid and its cohenrent nerve is equivalent to $S^{1}$.
The finial result is that $HOM(|\mathfrak{C}[k(S^{1})]|, C[C^{-1}])$ is a model for $\Lambda C$, since the nerve of $HOM(|\mathfrak{C}[k(S^{1})]|, C[C^{-1}])$ is equivalent to $ map(|\mathfrak{C}[k(S^{1})]|, C[C^{-1}])\sim Map(S^{1},N_{\bullet}C[C^{-1}])\sim Map(S^{1}, N_{\bullet} C)=\Lambda N_{\bullet}C$

N.B. The only point that I did not explained is the construction of $C[C^{-1}]$. Let 1 be the category with two objets a and b and a unique morphisms from $: f:a\rightarrow b$. Let $\widehat{1}=|\mathfrak{C}[k(\Delta^{1})]|$, then $ C[C^{-1}]$ is the pushout $colim (\sqcup_{mor C} \widehat{1}\leftarrow \sqcup_{mor C} 1\rightarrow C )$ i.e., for each porphism of $C$ there is a map $1\rightarrow C$.