In this paper we extend hyperparameter-free sparse signal reconstruction approaches to permit the high-resolution time delay estimation of spread spectrum signals and demonstrate their feasibility in terms of both performance and computation complexity by applying them to the ISO/IEC 24730-2.1 real-time locating system (RTLS). Numerical examples show that the sparse asymptotic minimum variance (SAMV) approach outperforms other sparse algorithms and multiple signal classification (MUSIC) regardless of the signal correlation, especially in the case where the incoming signals are closely spaced within a Rayleigh resolution limit. The performance difference among the hyperparameter-free approaches decreases significantly as the signals become more widely separated. SAMV is sometimes strongly influenced by the noise correlation, but the degrading effect of the correlated noise can be mitigated through the noise-whitening process. The computation complexity of SAMV can be feasible for practical system use by setting the power update threshold and the grid size properly, and/or via parallel implementations.