$\DeclareMathOperator\Spec{Spec}\DeclareMathOperator\Aut{Aut}$I think the issue here is that points of algebraic stacks have nontrivial automorphisms. In your example, the issue is that the 2 isomorphism is defined over $L$ but not over $K$, i.e., the automorphism group of that point change its structure upon a field extension. Certainly, for schemes, the map $\Spec L \to \Spec K$ is an epimorphism, since the automorphism group is trivial so it's defined over the integer. Now, if I want to remove that issue, I may want to assume that the automorphism group bears a trivial action of $\Aut_K(L)$. For nice algebraic stacks, the stabilizer groups are algebraic groups, so if $K$ is algebraically closed, then its structure is already stable under any field extension, i.e., its structure will not change anymore upon field extensions, this reduce the problem to the case where $L/K$ is algebraic.
Also, note that in the general case, where the source is no longer the spectrum of a field but only an algebraic stacks, then we can test whether two morphisms are the same on smooth presentations of the source algebraic stacks, and this reduces the problem to the case where the source is an affine scheme. Then we need to study the Galois action on both the set of 1-morphisms and also 2-morphisms. I guess that is the rough idea.
