The realization of Dirac criticality beyond the conventional Gross-Neveu-Yukawa (GNY) paradigm has become a major frontier in condensed matter physics. In this work, we introduce an SO(3)-symmetric bilayer SLAC fermion model with tunable inter-layer interactions that exhibits a rich quantum phase diagram. As the interaction strength increases, the system undergoes two distinct phase transitions. The primary transition is a continuous boundary separating a Dirac semimetal (DSM) from an SO(3)-broken ordered phase. Crucially, this transition evades the standard GNY universality class because the emergent order only gaps out a subset of the itinerant fermions. Using large-scale quantum Monte Carlo (QMC) simulations, we establish that this transition belongs to the Gross-Neveu-SO(3) universality class with N=6 irreducible Dirac cones and precisely extract the corresponding critical exponents. At stronger couplings, a second transition drives the system into an inter-layer SO(3)-symmetric superconducting (SC) state. We provide strong numerical evidence that this transition is first-order. Our study provides new insights into the exploration of Dirac criticality beyond the standard GNY universality class, and also offers a distinct platform for investigating SO(3)-symmetric superconductivity.