There is an increasing demand for Internet of Things (IoT) networks consisting of resource-constrained devices executing increasingly complex applications. Due to these resource-constraints, IoT devices will not be able to execute expensive tasks. One solution is to offload expensive tasks to resource-rich edge nodes. Which requires a framework that facilitates the selection of suitable edge nodes to perform task offloading. Therefore, in this paper, we present a novel trust model-driven system architecture, based on behavioural evidence, that is suitable for resource-constrained IoT devices that supports computation offloading. We demonstrate the viability of the proposed architecture with an example deployment of the Beta Reputation System trust model on real hardware to capture node behaviours. The open environment of edge-based IoT networks means that threats against edge nodes can lead to deviation from expected behaviour. Hence, we perform a threat modelling to identify such threats. The proposed system architecture includes threat handling mechanisms that provide security properties such as confidentiality, authentication and non-repudiation of messages in required scenarios and operate within the resource constraints. We evaluate the efficacy of the threat handling mechanisms and identify future work for the standards used.