To seek the possibility of baryonic superfluidity persisting at high density in neutron stars, the ^3 P_2-dominant superfluidity under developed neutral-pion (π^0) condensation, i.e., the coexistence of two hadronic condensates, is investigated by taking into account the Δ; isobar degrees of freedom. The problem is formulated as the ^3 P_2-dominant pairing of the quasi-neutrons, described by a superposition of the neutron and the Δ^0, in the alternating-layer-spin (ALS) structure equivalent to the π^0 condensation. Formulation given in previous papers is extended to thatexplicitly including Δ-mixing on the basis of the SU(4) quark model. Effects of Δ-mixing to the pairing potential are expressed in a compact form as a modification of the spin-isospin terms of nuclear forces. We find the possibility that the coexistence is likely up to the density region higher than 6ρ_0 (ρ;_0 being the nuclear density) in neutron star interior, with the superfluid critical temperature much higher than the internal temperature of neutron stars, if a type of nuclear forces with a gentle odd-state core is adopted. The conclusion depends on the features of the spin-isospin terms of nuclear forces enhanced by Δ-mixing, especially at short distances. for a type of nuclear forces with a steep odd-state core, the existence of the superfluid is limited to the density region less than about 4ρ_0. Discussion is given concerning the dependence of the results on the interaction, the parameters of the ALS structure, effective mass and so forth.