Comparison between analytic etale cohomology and algebraic etale cohomology for affinoids
Yes, there is such a comparison theorem. It works not only for finitely generated $K$-algebras, but more generally for finitely generated $A$-schemes for $A$ an affinoid algebra. This is done by Berkovich in the paper you mentioned (IHES 93), for coefficients in a ring whose torsion is prime to the residue characteristic.
He has extended those results later in a paper published in Israel Journal of Maths. The torsion (of the coefficients) is now allowed to be prime to the characteristic of the ground field. But I do not remember whether this holds for finitely generated schemes over an arbitrary affinoid algebra, or only over a field.
Not sure if this is still relevant, but the desired comparison in the case of "principal interest" to you can be proved in two different ways.
On one hand, you can deduce it from some recent results of Achinger, which imply that both cohomologies in question are unchanged if you replace the etale sites by the finite etale sites. Since the finite etale sites of $X$ and $X^{an}$ are canonically equivalent, this gives what you want. This works for any affinoid $\mathcal{A}$ over a complete discretely valued $K$ and any finite locally constant sheaf of abelian groups $\mathcal{L}$ on $X_{et}$.
Alternatively, and now for $K$ any complete nonarchimedean field, you can make a devissage to the case where $\mathcal{L}$ is a constant sheaf of finite abelian groups, which then is handled by Corollary 3.2.3 in Huber's book. To do this, pick a finite etale Galois cover $f:X'\to X$ with Galois group $G$ such that $f^{\ast}\mathcal{L}$ is constant, and use the Hochschild-Serre spectral sequences $H^i(G,H^j_{et}(X',f^{\ast}\mathcal{L})) \Rightarrow H^{i+j}_{et}(X,\mathcal{L})$ and $H^i(G,H^j_{et}(X'^{an},f^{an,\ast}\mathcal{L}^{an})) \Rightarrow H^{i+j}_{et}(X^{an},\mathcal{L}^{an})$. There is a canonical morphism from the first spec. seq. to the second, and it's an isomorphism on all terms of the $E_2$-page (by Huber's Corollary 3.2.3 plus some nonsense about compatibilities), so it gives an isomorphism on the abutments.