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Gravitational-wave (GW) detections of electromagnetically bright compact binary coalescences can provide an independent measurement of the Hubble constant $H_0$. In order to obtain a measurement that could help arbitrate the existing tension on $H_0$, one needs to fully understand any source of systematic biases for this approach. In this study, we aim to understand the impact of instrumental calibration errors (CEs) on the measurements of the luminosity distance, $D_L$, and the inferred $H_0$ value. We simulate binary neutron star mergers (BNSs), as detected by a network of Advanced LIGO and Advanced Virgo interferometers at their design sensitivity. We artificially add CEs equal to exceptionally large values experienced in LIGO-Virgo's third observing run (O3). We find that for individual BNSs at a network signal-to-noise ratio of 50, the systematic errors on $D_L$ - and hence $H_0$ - are still smaller than the statistical uncertainties. The biases become more significant when we combine multiple events to obtain a joint posterior on $H_0$. In the rather unrealistic case that the data around each detection is affected by the same CEs corresponding to the worst offender of O3, the true $H_0$ value would be excluded from the 90% credible interval after we combine $\sim$40 sources. If instead, 10% of the sources suffer from severe CEs, the true value of $H_0$ is included in the 90% credible interval even after 100 sources.