Transient contact responses in confined guide channels contain sharp events and parameter-dependent phase shifts, which make fixed-output full-history surrogates difficult to train. This study develops a time-conditioned residual surrogate (TC-ResNet) to propagate uncertainty in the bounded center-of-mass eccentricity components into the transient acceleration responses of a generic pressure-driven, high-overload axisymmetric body. The nonlinear reference model includes prescribed base pressure, wall contact and impact, velocity-dependent friction, gravity, pitch and yaw, and eccentricity. TC-ResNet predicts one response value for each parameter–time query by combining normalized physical parameters with Fourier-embedded time. The axial acceleration is modeled directly, whereas low-frequency radial trends and sliding root-mean-square (RMS) curves represent dominant lateral motion and local vibration intensity. On a common 108-case test set, TC-ResNet achieved the highest coefficient of determination (R²) for a_x^low (0.850), a_x^rms (0.839), and a_z^low (0.801), as well as the highest macro-mean R² (0.848). The fixed-output multilayer perceptron (MLP) remained best for a_y (0.995) and a_z^rms (0.775), demonstrating that the proposed model is not uniformly superior. Interval analysis shows that the prescribed pressure load limits axial sensitivity, whereas radial offsets alter eccentric pressure moments and wall contact; the axial offset primarily changes contact and friction moment arms, and inclination mainly affects the later radial response through gravity decomposition and accumulated contact differences. Accuracy approaches a plateau between 378 and 504 training cases. On the same central processing unit (CPU), TC-ResNet requires 0.081 s per curve (approximately 170× faster than the reference solver), and graphics processing unit (GPU) inference requires 0.019 s per curve. The reported envelopes support qualitative sensitivity analysis, but experimental calibration and validation remain necessary.