Real-time event detection and targeted decision making for emerging mission-critical applications require systems that extract and process relevant data from IoT sources in smart spaces. Oftentimes, this data is heterogeneous in size, relevance, and urgency, which creates a challenge when considering that different groups of stakeholders (e.g., first responders, medical staff, government officials, etc) require such data to be delivered in a reliable and timely manner. Furthermore, in mission-critical settings, networks can become constrained due to lossy channels and failed components, which ultimately add to the complexity of the problem. In this paper, we propose PrioDeX, a cross-layer middleware system that enables timely and reliable delivery of mission-critical data from IoT sources to relevant consumers through the prioritization of messages. It integrates parameters at the application, network, and middleware layers into a data exchange service that accurately estimates end-to-end performance metrics through a queueing analytical model. PrioDeX proposes novel algorithms that utilize the results of this analysis to tune data exchange configurations (event priorities and dropping policies), which is necessary for satisfying situational awareness requirements and resource constraints. PrioDeX leverages Software-Defined Networking (SDN) methodologies to enforce these configurations in the IoT network infrastructure. We evaluate our approach using both simulated and prototype-based experiments in a smart building fire response scenario. Our application-aware prioritization algorithm improves the value of exchanged information by 36% when compared with no prioritization; the addition of our network-aware drop rate policies improves this performance by 42% over priorities only and by 94% over no prioritization.