Is there a characterization of injective $C(X)$-modules analogous to Serre-Swan?

This is not really an answer. Rather, it deals with a different question, namely the analogue of your question in algebraic geometry. Perhaps it is still of some interest.

In algebraic geometry, one has an analogue of Serre--Swan, in which f.g. projective modules over a ring (commutative, with 1) $A$ correspond to finite rank locally free sheaves on Spec $A$.

If I recall correctly, under some assumptions, one can obtain injective $A$-modules as local cohomology sheaves of the structure sheaf supported at (not necessarily closed) points of Spec $A$.

E.g. if $A = k[x]$ (with $k$ an algebraically closed field), then the field of rational functions $k(x)$ is obtained as the local cohomology sheaf $\mathcal H^0_{\eta}(\mathcal O)$, where $\eta$ is the generic point of Spec $A$, while for a closed point $(x-a)$, the injective module $k[x,1/(x-a)]/k[x]$ is obtained as the local cohomology sheaf $\mathcal H^1_{(x-a)}(\mathcal O).$

This construction is due to Grothendieck, I think, and is discussed in Hartshorne's book Residues and duality, as a way of constructing canonical injective resolutions.

E.g. when $A = k[x]$, we have the injective resolution $$0 \to k[x] \rightarrow k(x) \to \bigoplus_{a \in k} k[x,1/(x-a)]/k[x] \to 0,$$ which can be rewritten in terms of sheaves on $X = $ Spec $A$ as $$0 \to \mathcal O_X \to \mathcal H^0_{\eta}(X,\mathcal O_X) \to \bigoplus_{a \in k} \mathcal H^1_{(x-a)}(X,\mathcal O_X) \to 0;$$ so the structure sheaf has an injective resolution whose $p$th term involves a sum over local cohomology sheaves supported at codimension $p$ points.

I don't have much of a feeling as to whether one can carry this over in any useful way to the topological setting, since the geometry of closed subsets in algebraic geometry is much more rigid than in general topology.