Let me try one more time, inspired by Pietro's idea.

Let's for now focus on $t=1/N$ and an interval $I_N$ of length $N$ located near $k\simeq (N/3)\log N$. Let's take $\lambda_k=(1\pm\delta)k$ on this interval, with $\delta=1/\log N$, and the sign is the same as that of $c_k$. Then the terms of
$$
\sum_{k\in I_N} c_k(e^{-\lambda_k t}- e^{-kt}) = \sum e^{-k/N}c_k(e^{\pm\delta k/N}-1) \quad\quad\quad\quad (1)
$$
are non-negative. In fact, let's also take $c_k$ with alternating signs. Then the sum is $\gtrsim N^{-1/3} \sum |c_k|$. We can easily make this large, for example by giving $|c|$ the constant value $|c_k|=N^{-1/2-\epsilon}$ on $I_N$ (note that this will keep $\sum_{I_N} |c_k|^2$ small, as required by the $\ell^2$ condition).

Now we define the whole sequence by first choosing $N_j$'s that increase very rapidly, and then defining $c_k$ as above on each of these intervals, and $c_k=0
$ otherwise. Notice that for $t=1/N_j$, if $k$ is taken from one of the other intervals $I_m$, $m\not= j$, then $e^{-\lambda_kt}-e^{-kt}$ is extremely small, because $tk$ is either very small or very large. So these intervals make negligible contributions to (1).

It follows that (1) does not go to zero as $t\to 0+$ along the sequence $t=1/N_j$, and as discussed earlier, this means that we have a counterexample.

Finally, an obvious modification also gives examples where $c\in\ell^p$ for any (or all) $p>1$.