The influence of confinement on the topological phases of superfluid 3He is studied using the torsional pendulum method. We focus on the phase transition between the chiral A-phase and the time-reversal-invariant B-phase, motivated by the prediction of a spatially-modulated (stripe) phase at the A-B phase boundary. We confine superfluid 3He to a single 1.08 μm thick nanofluidic cavity incorporated into a high-precision torsion pendulum, and study the pressure dependence of the phase diagram between 0.1 and 5.6 bar. We observe only small supercooling of the A-phase, in comparison to bulk or when confined in aerogel. This has a non-monotonic pressure dependence, suggesting that a new intrinsic B-phase nucleation mechanism operates under confinement, mediated by the putative stripe phase. Both the pressure dependence of the phase diagram and the relative superfluid fraction of the A and B phases, show that strong coupling is present at all pressures, with implications for the stability of the stripe phase.