Energy-efficient signal processing systems require estimation methods operating on data collected with low-complexity devices. Using analog-to-digital converters (ADC) with $1$-bit amplitude resolution has been identified as a possible option in order to obtain low power consumption. The $1$-bit performance loss, in comparison to an ideal receiver with $\infty$-bit ADC, is well-established and moderate for low SNR applications ($2/π$ or $-1.96$ dB). Recently it has been shown that for parameter estimation with state-space models the $1$-bit performance loss with Bayesian filtering can be significantly smaller ($\sqrt{2/π}$ or $-0.98$ dB). Here we extend the analysis to Bayesian smoothing where additional measurements are used to reconstruct the current state of the system parameter. Our results show that a $1$-bit receiver performing smoothing is able to outperform an ideal $\infty$-bit system carrying out filtering by the cost of an additional processing delay $Δ$.