= Solution
Interpret the additional copies in $Z_N$ as <independent> of the initial $T_1,\ldots,T_k$, as well as of one another. Set $r=k\lambda$. The <Cramér theorem> for rate-$r$ <exponential random variables> gives the <rate function>
$$
K_r(z)=\begin{cases}rz-1-\log(rz),&z>0,\\+\infty,&z\leq0.\end{cases}
$$
Indeed their <cumulant-generating function> is $\log(r/(r-\theta))$ for $\theta<r$, and the maximizer in its <Legendre-Fenchel transform> is $\theta=r-1/z$. For $Z_N/N$, replacing $N$ summands by $N-k$ does not change either the speed $N$ or this <rate function>. One can also see this directly from $N^{-1}\log\mathbb E e^{\theta Z_N}\to\log(r/(r-\theta))$ and the <Gärtner–Ellis theorem>.
The <product large-deviation principle>, followed by the <contraction principle for large deviations> under addition, now gives
$$
I_k(x)=\inf_{\substack{y\geq0,\ z>0\\y+z=x}}\{\lambda y+K_r(z)\}.
$$
For $x>0$ this is the minimum over $0<z\leq x$ of
$$
\lambda x+(r-\lambda)z-1-\log(rz).
$$
If $k>1$, its derivative with respect to $z$ is $r-\lambda-1/z$. Hence the optimum is $z=\min(x,1/[\lambda(k-1)])$, and
$$
\boxed{I_k(x)=
\begin{cases}
+\infty,&x\leq0,\\
k\lambda x-1-\log(k\lambda x),&0<x\leq\frac1{\lambda(k-1)},\\
\lambda x+\log\frac{k-1}{k},&x\geq\frac1{\lambda(k-1)}.
\end{cases}\qquad\text{speed }N.}
$$
The two expressions agree at the transition and have the same derivative $\lambda$. The minimum rate zero occurs at the typical value $x=1/(k\lambda)$. If $k=1$, the minimand is $\lambda x-1-\log(\lambda z)$, decreasing in $z$, so instead
$$
\boxed{I_1(x)=\lambda x-1-\log(\lambda x)\ (x>0),\qquad I_1(x)=+\infty\ (x\leq0).}
$$
These are <good rate functions>. The affine branch for $k>1$ is the <slow exponential plus an exponential sample mean> mechanism: the bulk ceases to carry additional excess, which instead comes from $T_1$. In particular, blindly applying the full <Gärtner–Ellis theorem> to the combined sum would be unjustified. Its limiting <cumulant-generating function> is $\log(r/(r-\theta))$ only for $\theta<\lambda$, with a finite boundary slope when $k>1$; <essential smoothness> fails. The product and contraction argument proves the missing lower bounds.
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