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All 21-cm signal experiments rely on electronic receivers that affect the data via both multiplicative and additive biases through the receiver's gain and noise temperature. While experiments attempt to remove these biases, the residuals of their imperfect calibration techniques can still confuse signal extraction algorithms. In this paper, the fourth and final installment of our pipeline series, we present a technique for fitting out receiver effects as efficiently as possible. The fact that the gain and global signal, which are multiplied in the observation equation, must both be modeled implies that the model of the data is nonlinear in its parameters, making numerical sampling the only way to explore the parameter distribution rigorously. However, multi-spectra fits, which are necessary to extract the signal confidently as demonstrated in the third paper of the series, often require large numbers of foreground parameters, increasing the dimension of the posterior distribution that must be explored and therefore causing numerical sampling inefficiencies. Building upon techniques in the second paper of the series, we outline a method to explore the full parameter distribution by numerically sampling a small subset of the parameters and analytically marginalizing over the others. We test this method in simulation using a type-I Chebyshev band-pass filter gain model and a fast signal model based on a spline between local extrema. The method works efficiently, converging quickly to the posterior signal parameter distribution. The final signal uncertainties are of the same order as the noise in the data., Comment: 15 pages, 7 figures, submitted to ApJ