We say $(M,N)$ is a $C^k$ pair of manifolds if $M$ and $N$ are $C^k$ manifolds and we implicitly fix an $C^k$ embedding $i: N \hookrightarrow M$ so that we identify $N$ with its image in $M$. A classical result of differential topology is that any $C^k$ manifold for $k \geq 1$ admits a unique real analytic structure. (See, for example, the MO discussion.)
My question is whether an existence and uniqueness result for pairs $(M,N)$ in the following sense is true? Let $M^a$ and $N^a$ denote the (unique) $C^\omega$-structure on $M$ and $N$. Then, there exists a $C^\infty$ isotopy $\phi_t$ on $M$, $t \in [0,1]$, such that $\phi_0 = id_M$ and $N^a \cong \phi_1(N) \hookrightarrow M^a$ is $C^\omega$. For any two such $C^\infty$ isotopies $\phi_t$, $\phi_t^\prime$, there is a $C^w$ isotopy $\psi_s$ on $M^a$, $s \in [0,1]$ such that $\psi_1 ( \phi_1(N)) = \phi_1^\prime(N)$.
In simple words, can the smoothing of $N$ be made in $M$ and is the resulting smoothing unique relative to $M$?