We explore the properties of spatiotemporal solitons within nonlinear graded-index (GRIN) optical fibers, specifically addressing their formation, stability, and breathing dynamics. We solve the (3+1)-dimensional nonlinear Schrödinger equation employing a semi-analytical variational approach, which leads to a set of reduced coupled ordinary differential equations describing the bistable bullet dynamics. The stability of these states is assessed using the Vakhitov-Kolokolov criterion and linear stability analysis within the Kantorovich optimization framework. The breathing dynamics of the optical bullets are investigated adopting the potential formalism, revealing an oscillatory behavior and self-imaging dynamics in GRIN fibers. The analytical results demonstrate strong agreement with full numerical simulations, validating the robustness of our theoretical model. Our research enhances the understanding of stabilizing self-repeated 3D optical structures, which is beneficial for the development of advanced photonic technologies.