The non-ideal sparse representation (SR) and harmonics are universal problems in linear frequency modulated (LFM) signal recovery from one-bit data contaminated by noise. Given these issues, we develop a consistency algorithm with the one-sided weighted quadratic penalty (OWQP) to reconstruct LFM signals quantized to one bit. In our recovery model, a stochastic quantization approach is first employed to address the harmonic problem in one-bit sampling. Then, depending on the block-shaped fractional Fourier spectrum of LFM signals and the symmetry of the sinc function, we extend the sparse support set of signals to decrease the SR error caused by the non-ideal SR of LFM signals in the fractional Fourier transform basis. After identifying this set, the original under-determined optimization is readily reformulated as an overdetermined one estimating a proxy of the desired signal. An OWQP function is further proposed to penalize the noise-induced inconsistencies of the proxy with measurements. Numerical results show that the performance of our algorithm is superior to state-of-the-art methods in terms of the reconstructed signal-to-noise ratios and mean square errors.