A bandwidth-controlled antiferromagnetic Mott-insulating phase in Ca₃(Ru₁₋ₓTiₓ)₂O₇ is realized through isovalent substitution at the Ru site. For a dilute substitution with only 1% Ti, the Mott insulator ground state remains nearly degenerate with the ground state of pristine Ca₃Ru₂O₇, where the Ru moments are ferromagnetically aligned within the metallic RuO₂ bilayers stacked in an antiferromagnetic fashion. The exceptionally shallow free energy landscape of this doped compound arises from intertwined electron-electron and electron-lattice interactions. This makes its magnetic and transport properties highly sensitive to external perturbations. We systematically investigated magnetic-field-induced phase switching in Ca₃(Ru_(0.99)Ti_(0.01))₂O₇ to explore its magnetic H-T phase diagram. With the field applied along the easy b-axis, parallel to the antiferromagnetic moments, the magnetization exhibits a first-order spin-flop transition at ≈ 6 T, indicating reorientation of the Ru moments perpendicular to the field. The transition is accompanied by a decrease in the electrical resistance, but the spin-flop phase remains insulating. Above 10.5 T, all Ru moments align with the b-axis, resulting in a forced ferromagnetic metallic phase. In contrast, neither spin-flop nor forced-ferromagnetic phases are observed up to 14 T, when the field is applied along the a-axis. While the electronic kinetic energy and the electron-lattice coupling contribute to the free-energy balance of this system, the resulting H-T phase diagram is remarkably simple and closely resembles that of a canonical anisotropic antiferromagnet, albeit with substantially renormalized critical fields.