= Solution
With the other coordinates fixed, the smooth part of the objective has derivative at $\beta_1=0$
$$
\frac1n\left(\kappa_1-(\kappa_2-\kappa_1)\right)
=\frac{2\kappa_1-\kappa_2}{n}.
$$
The subdifferential of the $\ell^1$ penalty at zero is $\gamma[-1,1]$. Therefore
$$
\partial_{\beta_1}f(0,\beta_2,\ldots,\beta_p)
=\frac{2\kappa_1-\kappa_2}{n}+\gamma[-1,1].
$$
The stated condition $|2\kappa_1-\kappa_2|\leq\gamma$ is stronger than the needed condition $|2\kappa_1-\kappa_2|\leq n\gamma$, and hence places zero in this subdifferential. The one-variable exponential loss is strictly convex because the mismatch set is nonempty, so the minimizer is unique:
$$
\boxed{\operatorname{argmin}_{\beta_1\in\mathbb R}f(\beta_1,\beta_2,\ldots,\beta_p)=\{0\}}.
$$
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