I can't solve your sign question, but just a minor correction: should $\kappa$ be playing a role in the definition, in particular controlling the indexing set for the $\underline i$?

@Darij: the dust has now settled: arxiv.org/abs/1309.6467

Is it obvious that you should avoid midspheres if you want a 3-dimensional Poncelet's porism? It feels to me (based on zero examples) that with just an insphere and a circumsphere you have too much freedom in choosing your polyhedron.

While I'm generally reluctant to make a comment that says "you're asking the wrong question" ... why are you asking for polyhedra with just an insphere and a circumsphere? It seems natural (to me) to ask for your polyhedron to have a midsphere (i.e. a sphere tangent to every edge) as well. You then have the nice inductive property that every face (and every vertex figure?) of a "trispherical" polyhedron is a bicentric polygon.

@David & @Philippe: Great answers; many thanks.

@Marc: I meant lowest. There are two competing conventions, in which the same terminology is used. The reference you cite considers $p$-regular partitions rather than $p$-restricted partitions, so everything has to be transposed from there, and in particular, highest becomes lowest. For example, for the partition $(2,1,1)$, both the addable $0$-nodes $(1,3)$ and $(2,2)$ are conormal, and we add the lower one to get the partition $(2,2,1)$; if we instead added the higher conormal node, we'd obtain $(3,1,1)$, which is not $2$-restricted.

@Christian: thanks for doing this - that's a good resource.