Threshold Fully Homomorphic Encryption (FHE) enables arbitrary computation on encrypted data, while distributing the decryption capability across multiple parties. A primary application of interest is low-communication multi-party computation (MPC), which benefits from a fast and secure threshold FHE decryption protocol. Several works have addressed this problem, but all existing solutions rely on "noise flooding" for security. This incurs significant overhead and necessitates large parameters in practice, making it unsuitable for many real-world deployments. Some constructions have somewhat better efficiency, but at the cost of weaker, non-simulation-based security definitions, which limits their usability and composability. In this work, we propose a novel threshold FHE decryption protocol that avoids "noise flooding" altogether, and provides strong simulation-based security. Rather than masking the underlying ciphertext noise, our technique securely removes it via an efficient MPC rounding procedure. The cost of this MPC is mitigated by an offline/online design that preprocesses special gates for secure comparisons in the offline phase, and has low communication and computation in the online phase. This approach is of independent interest, and should also benefit other MPC protocols (e.g., secure machine learning) that make heavy use of non-linear comparison operations. We prove our protocol secure in the Universal Composability (UC) framework, and it can be generally instantiated for a variety of adversary models (e.g., security-with-abort against a dishonest majority, or guaranteed output delivery with honest majority). Compared to the state of the art, our protocol offers significant gains both in the adversary model (i.e., dishonest vs. honest majority) and practical performance: empirically, our online phase obtains approximately 20,000x better throughput, and up to a 37x improvement in latency.