@user509184 That would be a great start! If I remember correctly, the proof is pretty similar (up to throwing some assumptions on M) and would probably suffice

The shortest definition would be the derived functors of the functor taking a representation to its invariants, though it can also be defined in terms of a cochain complex

@Eoin Proposition 3.3.14 gives a criterion for extending a section to first order deformation, so for $A=k[\varepsilon]/(\varepsilon)^2$, not a general infinitesimal or formal deformation. Maybe it is possible to extend the argument. I just hoped a condition might be known already

Ah yes, you are right. I was thinking of complex deformations when formulating the question.

I think this is certainly along the lines of what I'm looking for. I'll have to read it in more detail, but I very much appreciate the suggestion

I don't quite know if there is a canonical way, but the obvious thing would be that if, for every pair of objects $M,N$,$\omega_M(X)\cong\omega_N(X)$ for every object in both $\langle M\rangle$ and $\langle N\rangle$. Which seems something like a "global section of the sheaf of functors to vector spaces", in a non rigourous sense. Then in order to be canonical, you would want the isomorphism $\omega_M(X)\cong\omega_N(X)$ to be canonical in some sense, which would probably be classified by some sort of cohomology-like object