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Many recent works have shown substantial efficiency boosts from performing inference tasks on Internet of Things (IoT) nodes rather than merely transmitting raw sensor data. However, such tasks, e.g., convolutional neural networks (CNNs), are very compute intensive. They are therefore challenging to complete at sensing-matched latencies in ultra-low-power and energy-harvesting IoT nodes. ReRAM crossbar-based accelerators (RCAs) are an ideal candidate to perform the dominant multiplication-and-accumulation (MAC) operations in CNNs efficiently, but conventional, performance-oriented RCAs, while energy-efficient, are power hungry and ill-optimized for the intermittent and unstable power supply of energy-harvesting IoT nodes. This paper presents the ResiRCA architecture that integrates a new, lightweight, and configurable RCA suitable for energy harvesting environments as an opportunistically executing augmentation to a baseline sense-and-transmit battery-powered IoT node. To maximize ResiRCA throughput under different power levels, we develop the ResiSchedule approach for dynamic RCA reconfiguration. The proposed approach uses loop tiling-based computation decomposition, model duplication within the RCA, and inter-layer pipelining to reduce RCA activation thresholds and more closely track execution costs with dynamic power income. Experimental results show that ResiRCA together with ResiSchedule achieve average speedups and energy efficiency improvements of 8× and 14× respectively compared to a baseline RCA with intermittency-unaware scheduling.