The combination of Integrated Sensing and Communication (ISAC) and Mobile Edge Computing (MEC) enables devices to simultaneously sense the environment and offload data to the base stations (BS) for intelligent processing, thereby reducing local computational burdens. However, transmitting raw sensing data from ISAC devices to the BS often incurs substantial fronthaul overhead and latency. This paper investigates a three-tier collaborative inference framework enabled by Integrated Sensing, Communication, and Computing (ISCC), where cloud servers, MEC servers, and ISAC devices cooperatively execute different segments of a pre-trained deep neural network (DNN) for intelligent sensing. By offloading intermediate DNN features, the proposed framework can significantly reduce fronthaul transmission load. Furthermore, multiple-input multiple-output (MIMO) technology is employed to enhance both sensing quality and offloading efficiency. To minimize the overall sensing task inference latency across all ISAC devices, we jointly optimize the DNN partitioning strategy, ISAC beamforming, and computational resource allocation at the MEC servers and devices, subject to sensing beampattern constraints. We also propose an efficient two-layer optimization algorithm. In the inner layer, we derive closed-form solutions for computational resource allocation using the Karush-Kuhn-Tucker conditions. Moreover, we design the ISAC beamforming vectors via an iterative method based on the majorization-minimization and weighted minimum mean square error techniques. In the outer layer, we develop a cross-entropy based probabilistic learning algorithm to determine an optimal DNN partitioning strategy. Simulation results demonstrate that the proposed framework substantially outperforms existing two-tier schemes in inference latency.