This paper proposes a low-computational Bayesian algorithm for noisy sparse recovery (NSR), called BHT-BP. In this framework, we consider an LDPC-like measurement matrices which has a tree-structured property, and additive white Gaussian noise. BHT-BP has a joint detection-and-estimation structure consisting of a sparse support detector and a nonzero estimator. The support detector is designed under the criterion of the minimum detection error probability using a nonparametric belief propagation (nBP) and composite binary hypothesis tests. The nonzeros are estimated in the sense of linear MMSE, where the support detection result is utilized. BHT-BP has its strength in noise robust support detection, effectively removing quantization errors caused by the uniform sampling-based nBP. Therefore, in the NSR problems, BHT-BP has advantages over CS-BP which is an existing nBP algorithm, being comparable to other recent CS solvers, in several aspects. In addition, we examine impact of the minimum nonzero value of sparse signals via BHT-BP, on the basis of the results of the recent literature. Our empirical result shows that variation of x_min is reflected to recovery performance in the form of SNR shift.