Designing Doppler-resilient unimodular discrete phase-coded waveforms (DPWs) with low delay-Doppler sidelobes is critical for multiple-input multiple-output (MIMO) radar. Existing block coordinate descent (BCD) methods suffer from high computational cost for designing long sequences or large waveform sets. Meanwhile, learning-based alternatives such as the soft-quantization network (SQN) only address correlation optimization in the delay domain, without considering ambiguity function (AF) optimization in the joint delay-Doppler domain. To address these issues, this paper proposes a novel Doppler-resilient DPW design framework, termed SQNGD, for joint transmit-receive optimization that simultaneously optimizes the auto-AF, cross-AF (CAF), and signal-to-noise ratio loss (SNRL) under unimodular constraints. To solve the multi-objective optimization problem (MOOP), a joint transmit-receive design and an alternating optimization strategy are developed. The transmit waveforms are optimized via soft-quantization-based differentiable parameterization, while the receive filters are updated by gradient descent (GD) with an energy constraint and SNRL penalty. An FFT-accelerated evaluation of the AF and CAF is further incorporated, reducing the optimization time by 1.9x - 11x compared with the state-of-the-art (SOTA) majorization-minimization-coordinate descent (MMCD) method. Numerical results show that SQNGD achieves a peak sidelobe level (PSL) of approximately -43 dB over the Doppler range [-0.5,0.5] and -31 dB over [-600,600], respectively, outperforming MMCD by 5.85 dB and 3.45 dB, while maintaining the same SNRL of 0.5 dB.