This article seeks to advance coded compressed sensing (CCS) as a practical scheme for unsourced random access. The original CCS algorithm features a concatenated structure where an inner code is tasked with support recovery, and an outer tree code conducts message disambiguation. Recently, a link between CCS and sparse regression codes was established, leading to the application of approximate message passing (AMP) to CCS. This connection was subsequently strengthened by integrating AMP and belief propagation on the outer code through a dynamic denoiser. Along these lines, this work shows how block diagonal sensing matrices akin to those used in traditional CCS, together with the aforementioned dynamic denoiser, form an effective means to get good performance at low-complexity. This novel architecture can be used to scale this scheme to dimensions that were previously impractical. Findings are supported by numerical simulations.