In 6G networks, integrated sensing and communication (ISAC) is envisioned as a key technology that enables wireless systems to perform joint sensing and communication using shared hardware, antennas and spectrum. ISAC designs facilitate emerging applications such as smart cities and autonomous driving. Such applications also demand ultra-reliable and low-latency communication (URLLC). Thus, an ISAC-enabled URLLC system can prioritize time-sensitive targets and ensure information delivery under strict latency and reliability constraints. We propose a bi-static MIMO ISAC system to detect the arrival of URLLC messages and prioritize their delivery. In this system, a base station (BS) communicates with a user equipment (UE) and a sensing receiver (SR) is deployed to collect echo signals reflected from a target of interest. The BS regularly transmits messages of enhanced mobile broadband (eMBB) services to the UE. During each eMBB transmission, if the SR senses the presence of a target of interest, it immediately triggers the transmission of an additional URLLC message. To reinforce URLLC transmissions, we propose a dirty-paper coding (DPC)-based technique that mitigates the interference of both eMBB and sensing signals. To decode the eMBB message, we consider two approaches for handling the URLLC interference: treating interference as noise and successive interference cancellation. For this system, we formulate the rate-reliability-detection trade-off in the finite blocklength (FBL) regime by evaluating the communication rate of the eMBB transmissions, the reliability of the URLLC transmissions and the probability of the target detection. Our numerical analysis show that our proposed DPC-based ISAC scheme significantly outperforms power-sharing and traditional time-sharing schemes. In particular, it achieves higher eMBB transmission rate while satisfying both URLLC and sensing constraints.