We investigate the effects of diamagnetic doping in the solid-solution series Tb1-xYx(HCO2)3, in which the parent Tb(HCO2)3 phase has previously been shown to host a combination of frustrated and quasi-1D physics, giving rise to a triangular Ising antiferromagnetic ground state that lacks long range 3D order. Heat capacity measurements show three key features: (i) a low temperature Schottky anomaly is observed, which is constant as a function of x; (ii) the transition temperature and associated entropy change are both surprisingly robust to diamagnetic doping; and (iii) an additional contribution at T < 0.4 K appears with increasing x. The origin of this unusual behaviour is rationalised in terms of the fragmentation of quasi-1D spin chains by the diamagnetic Y 3+ dopant. Magnetocaloric measurements show a nonlinear dependence on x. The mass-weighted magnetocaloric entropy decreases across the series from the promising values in Tb(HCO2)3; however, the magnetocaloric entropy per magnetic Tb 3+ ion first decreases then increases with increasing x. Our results establish Tb1-xYx(HCO2)3 as a model system in which to explore the functional ramifications of dilution in a low-dimensional magnet.