Motivated by recent experimental progress on high-temperature superconductivity in bilayer nickelates, we investigate the phase diagram of the normal state in a bilayer Kondo lattice model using single-site dynamical mean-field theory (DMFT). When the interlayer tunneling t_⊥ is absent, we identify a non-Fermi-liquid (NFL) critical point tuned by the interlayer spin coupling J_⊥ or hole doping x, which separates a standard Fermi liquid in the overdoped region from a distinct pseudogap (PG) metal in the underdoped regime. This PG phase, which we term the `second Fermi liquid' (sFL), exhibits small hole pockets and violates the perturbative Luttinger theorem despite the absence of symmetry breaking or fractionalization. The PG metal behaves like a heavy Fermi liquid, with small quasi-particle residue and large effective mass. We also provide an intuitive analytical description of the pseudogap and the ground-state wave function based on an ancilla-fermion framework. Inside the PG phase, we interpret the ancilla fermion as a spin-polaron and demonstrate a Kondo-like resonance peak in the spectral function of this composite fermion directly in DMFT calculation. Extending the analysis to finite t_⊥, we apply this framework to the bilayer nickelate La₃Ni₂O₇. We propose that current experimental samples (x ≈ 0.5) reside in the overdoped FL regime, suggesting that the pseudogap phase and the NFL criticality may be accessed via electron doping.