Integrated sensing and communication (ISAC) provides an emerging paradigm for enabling a variety of next-generation wireless services and applications. Due to the limited computation resources on ISAC devices and the latency as well as the reliability requirements, we propose a paradigm of mobile edge computing (MEC) aided ISAC with short-packet transmissions, where multiple ISAC devices adopt short-packet transmissions to offload their sensed radar data to an edge-server for analysis. We adopt the mutual information to measure the performance of radar sensing and quantify the reliability and latency performances for analyzing the radar-data via edge computing. We formulate an energy minimization problem that jointly optimizes the size of each short packet, the duration of each short packet, the computing-capacity allocations of edge-server, the beamforming of the radar sensing and the offloading transmission, while providing guaranteed performances for the radar sensing, the latency for radar-data analysis, and the reliability of offloading transmission. We identify the hierarchical structure of the formulated problem and divide the problem into three subproblems. For both the bottom-layer problem optimizing the computing-capacity allocations of the edge-server and the middle-layer problem optimizing the size of each short packet and the duration of each short packet, we derive their solutions analytically. Finally, for the top-layer problem optimizing the beamforming of the radar sensing and the offloading transmission, we transform it into a difference of convex (DC) problem which can be efficiently solved. We show the performance advantages of our proposed scheme. The simulation results show that our proposed algorithm can outperform the benchmark algorithms.