We consider a lattice polymer model of the two-tolerant type (i.e., a random walk allowed to visit lattice bonds at most twice), in which doubly visited bonds yield an attractive energy term (pairing energy). Such a model has been previously proposed as a rough, nonspecific description of the RNA folding mechanism. Indeed, the model predicts, besides the usual theta collapse, an extra transition to a low-temperature fully paired state. In the current work, we propose an extension of the model, in which a "micromolecular" chemical species can bind the polymer and locally forbid segment pairing. We investigate equilibrium thermodynamics in the grand-canonical picture, at the level of a Bethe approximation, which is, a refined mean-field technique, equivalent to the exact solution on a random-regular graph. The general trend we observe is that expected from the mechanism implemented in the model (increasing micromolecule concentration favors unfolding and lowers the transition temperature), but the resulting phase diagram turns out to be remarkably interesting and rich.