The intramolecular exchange interactions within the single-molecule magnet (SMM) "butterfly" molecule [Fe₃Ln(μ₃-O) ₂(CCl₃COO)₈(H₂O)(THF)₃], where Ln(III) represents a lanthanide cation, are determined in a combined experimental [x-ray magnetic circular dichroism (XMCD) and vibrating sample magnetometer (VSM)] and theoretical work. Compounds with Ln=Gd and Dy, which represent extreme cases where the rare earth presents single-ion isotropic and uniaxial anisotropy, on one hand, and with Ln=Lu and Y(III) as pseudolanthanide substitutions that supply a nonmagnetic Ln reference case, on the other hand, are studied. The Dy single-ion uniaxial anisotropy is estimated from ab initio calculations. Low-temperature (T 2.5 K) hard x-ray XMCD at the Ln L _2,3 edges and VSM measurements as a function of the field indicate that the Ln moment dominates the polarization of the molecule by the applied field. Within the {Fe₃LnO₂} cluster the Ln-Fe₃ subcluster interaction is determined to be antiferromagnetic in both Dy and Gd compounds, with values J Dy-Fe₃=-0.4 K and J Gd-Fe₃=-0.25 K, by fitting to spin Hamiltonian simulations that consider the competing effects of intracluster interactions and the external applied magnetic field. In the uniaxial anisotropic {Fe₃DyO₂} case, a field-induced reorientation of the Fe₃ and Dy spins from an antiparallel to a parallel orientation takes place at a threshold field (μ₀H=4 T). In contrast, in isotropic {Fe₃GdO₂} this reorientation does not occur.