We present Accelerated Fourier SAT (AFSAT), a GPU-accelerated solver for pseudo-Boolean satisfiability based on continuous local search (CLS). AFSAT realises the proof-of-concept approach, FastFourierSAT, into a fully-engineered solver supporting any heterogeneous mixture of symmetric constraint types and lengths within a single problem instance. Using the JAX compiler, AFSAT leverages pure function composition, automatic vectorisation, automatic differentiation, and just-in-time (JIT) compilation to perform massively parallel CLS across batches of candidate assignments. We demonstrate substantially improved numerical stability, runtime performance, and memory efficiency over the proof-of-concept. We achieve this by way of identifying and addressing various limitations that arise from memory latency and floating-point representation, as well as leveraging automatic parallelisation and compact representations. The inherent representational and stability limitations of floating point are partially addressed by a tailored discrete Fourier transform implementation. We achieve near-linear throughput when scaling to multiple accelerators via JAX array sharding.