Scalable superconducting quantum computation requires fast, high-fidelity readout. Currently, this is assisted through cryogenic amplification of the readout signal. In traditional dispersive readout schemes, the amplifiers that are used to provide high signal-to-noise ratio also generate thermal noise that can decohere the qubits, demanding additional ferrite-based isolators to suppress this noise. Here, we integrate a chip-scale microwave switch, the tunable inductive bridge, into the readout chain of a fixed-frequency transmon. Flux-pulsing the switch, such that it transmits only during the readout, did not make a discernable difference to qubit coherence, residual excited-state population, or readout fidelity metrics. By injecting broadband noise that mimics amplifier thermal photons, we show that the switch suppresses the resulting qubit dephasing by more than 30~dB, consistent with its independently measured on-off ratio, establishing it as a potential chip-based integrated replacement for readout-chain isolators.