  We study the interplay of strong electron correlations and intra-triangle
spin exchange at two-thirds filling of the single-orbital Hubbard model on the
decorated honeycomb lattice using rotationally invariant slave bosons. We find
that the spin exchange tunes between a spin-$1$ Mott insulator, a metal, and a
spin-$0$ Mott insulator when the exchange is antiferromagnetic. The Mott
insulators occur from effective intra-triangle multi-orbital interactions and
are adiabatically connected to the ground state of an isolated triangle. An
antiferromagnetic spin exchange, as determined by the Goodenough-Kanamori
rules, may occur in coordination polymers from kinetic exchange via the
ligands. We characterize the magnetism in the regime where spin-triplets
dominate. For small $U$ a spin-$1$ Slater insulator occurs with
antiferromagnetic order between triangles. Magnetism in the spin-$1$ Mott
insulator is described by a spin-$1$ Heisenberg model on a honeycomb lattice,
whose ground state is N\'{e}el ordered.
