Knowing that CI is variational for any basis set, why is coupled-cluster variational in the full correlation limit?
The CI (configuration interaction) energy, given by Eqs. 3.9-3.10 of this, is an upper bound to the exact ground state energy (i.e. CI is variational). Coupled cluster energies are not generally variational, as explained in the above linked question, except in the case of CC($N$) for an $N$-electron system. The reason is because CC($N$) for an $N$-electron system is equivalent to FCI (full configuration interaction) which is an example of CI, and all CI energies are variational. 

Since I was asked for a publication as a reference, a good one is this, in which it is shown that CC and CI become the exact same in the full-correlation limit.