This study examines currency competition between the Dollar and Bitcoin, focusing on monetary policy. Using theory and lab experiments, we show that proportional transfers (interest on Dollar balances) operate in two regimes. Depending on inflation expectations, policy may raise Dollar trade shares and reduce velocity, weakening price pass-through and generating Bitcoin inflation spillovers despite its rigid supply. Alternatively, pass-through can be strong with little effect on trade. Expectations drive these self-fulfilling dynamics and endogenously shape Dollar price stickiness. Even with flexible prices, money need not be neutral. Our results reveal limits of central bank influence in pluralistic systems.