The rule sending G to this particular s.e.s. [0 to G to A to B to 0] is concrete enough that it should be possible to verify directly that it is exact in G, no pushouts neded? The remaining argument on differentials then makes sense (to me).

Thanks! To summarize what's behind that Mazur–Roberts reference, for the sake of anyone reading this post later on: Writing G* for the Cartier dual of G, one embeds G = Hom(G*,G_m) into A = Map(G*,G_m) (maps of S-schemes), and takes B to be the quotient. (The Map in question is representable and that A,B have the stated properties.)

Here is my current belief: I was executing Deligne's formula wrong, and if K=C then one should get two copies of -i(-1)^p, i.e. simply -1. At least this would be consistent with my expecting +/-1 in all cases, and some other computations I've done. I would still love an expert to weigh in on this, and especially on what happened to Fontaine–Perrion-Riou in the case K=R.

For complex CM fields, the your possible weights $\sigma_i/\overline\sigma_i$ are not the whole story: you gave a sublattice of index $2^d$. It's more that, for each CM type $\Phi$, i.e. a choice of one element of each $\{\sigma_i,\overline\sigma_i\}$, you can add the weight $\sum_{\sigma \in \Phi} \sigma$. (Thus your example is the difference of a CM type and its conjugate, or twice a CM type minus the norm.) Note that the Hecke characters attached to CM abelian varieties need precisely the weights $\Phi$.