Let $T$ be the theory of linear dense orders with no endpoints and let $\mathcal{L}=\{<,c_0,c_1,\dots\}$ be the language that consists of a binary relation symbol and a countable amount of constant symbols. Given $N\in\mathbb{N}$, I want to show that the $\mathcal{L}$-theory $T_N=T\cup\{c_n<c_{n+1}:0<n<N\}$ is $\omega$-categorical.
I know that $T$ is $\omega$-categorical, so my idea to prove this was the following: Let $\frak{A}$ and $\frak{B}$ be countably infinite models of $T_N$. Let $A$ and $B$ be the universes of $\frak{A}$ and $\frak{B}$, respectively. Define $\tilde{A}=A\setminus\{c_0^{\frak{A}},\dots, c_N^{\frak{A}}\}$ and $\tilde{B}=B\setminus\{c_0^{\frak{B}},\dots, c_N^{\frak{B}}\}$. $(\tilde{A},<)$ and $(\tilde{B},<)$ are both countably infinite models of $T$, so there exists an isomorphism $\psi$ from $(\tilde{A},<)$ to $(\tilde{B},<)$. Finally, define $\phi:A\to B$ by $\phi\restriction_\tilde{A}=\psi$ and $\phi(c_i^{\frak{A}})=c_i^\frak{B}$ for all $i\in\{0,\dots,N\}$. However, I can't seem to show that if $a\in\tilde{A}$ and $a<c_i^{\frak{A}}$, then $\phi(a)<c_i^{\frak{B}}$; so this may not be an isomorphism.
Is my approach correct by any chance? If not, how could I construct an isomorphism between these structures?