Key performance characteristics are demonstrated for the microwave SQUID multiplexer ($mu$MUX) coupled to transition edge sensor (TES) bolometers that have been optimized for cosmic microwave background (CMB) observations. In a 64-channel demonstration, we show that the $mu$MUX produces a white, input referred current noise level of 29~pA$/sqrt{mathrm{Hz}}$ at -77~dB microwave probe tone power, which is well below expected fundamental detector and photon noise sources for a ground-based CMB-optimized bolometer. Operated with negligible photon loading, we measure 98~pA$/sqrt{mathrm{Hz}}$ in the TES-coupled channels biased at 65% of the sensor normal resistance. This noise level is consistent with that predicted from bolometer thermal fluctuation (i.e., phonon) noise. Furthermore, the power spectral density exhibits a white spectrum at low frequencies ($sim$~100~mHz), which enables CMB mapping on large angular scales that constrain the physics of inflation. Additionally, we report cross-talk measurements that indicate a level below 0.3%, which is less than the level of cross-talk from multiplexed readout systems in deployed CMB imagers. These measurements demonstrate the $mu$MUX as a viable readout technique for future CMB imaging instruments.
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