Complex-oxide materials tuned to be near phase boundaries via chemistry/composition, temperature, pressure, etc. are known to exhibit large susceptibilities. Here, we observe a strain-driven nanoscale phase competition in epitaxially constrained Bi0.7La0.3FeO3 thin films near the antipolar-nonpolar phase boundary and explore the evolution of the structural, dielectric, (anti)ferroelectric, and magnetic properties with strain. We find that compressive and tensile strains can stabilize an antipolar PbZrO3-like Pbam phase and a nonpolar Pnma orthorhombic phase, respectively. Heterostructures grown with little to no strain exhibit a self-assembled nanoscale mixture of the two orthorhombic phases, wherein the relative fraction of each phase can be modified with film thickness. Subsequent investigation of the dielectric and (anti)ferroelectric properties reveals an electric-field-driven phase transformation from the nonpolar phase to the antipolar phase. X-ray linear dichroism reveals that the antiferromagnetic-spin axes can be effectively modified by the strain-induced phase transition. This evolution of antiferromagnetic-spin axes can be leveraged in exchange coupling between the antiferromagnetic Bi0.7La0.3FeO3 and a ferromagnetic Co0.9Fe0.1 layer to tune the ferromagnetic easy axis of the Co0.9Fe0.1. These results demonstrate that besides chemical alloying, epitaxial strain is an alternative and effective way to modify subtle phase relations and tune physical properties in rare earth-alloyed BiFeO3. Furthermore, the observation of antiferroelectric-antiferromagnetic properties in the Pbam Bi0.7La0.3FeO3 phase could be of significant scientific interest and great potential in magnetoelectric devices because of its dual antiferroic nature.