Co-orbital planets (in a 1:1 mean motion resonance) can be formed within a Laplace resonance chain. Here, we develop a secular model to study the dynamics of the resonance chain p:p:p+1, where the co-orbital pair is in a first-order mean motion resonance with the outermost third planet. Our model takes into account tidal dissipation through the use of a Hamiltonian version of the constant time-lag model, which extends the Hamiltonian formalism of the point-mass case. We show the existence of several families of equilibria, and how these equilibria extend to the complete system. In one family, which we call the main branch, a secular resonance between the libration frequency of the co-orbitals and the precession frequency of the pericentres has unexpected dynamical consequences when tidal dissipation is added. We report the existence of two distinct mechanisms that make co-orbital planets much more stable within the p:p:p+1 resonance chain rather than outside it. The first one is due to negative real parts of the eigenvalues of the linearized system with tides, in the region of the secular resonance mentioned above. The second one comes from non-linear contributions of the vector field and it is due to eccentricity damping.