The reference is page 171 of the online edition, the hardcopy has the same text here. Indeed, as Careful has pointed out, my example was not linear, and thus not in the algebraic dual, but what Rovelli does not spell out -- as I have learned from him (see below) -- is that the space K_diff he...
My advisor and I have stumbled upon a very strange claim in Rovelli's book. There he defines the P_diff map from S0 onto its algebraic dual S0* as
P_diff(Psi) Psi' = sum (Psi'' = phi Psi) (Psi'',Psi')
This is indeed a well-defined map that yields diff-invariant states. However, Rovelli claims...