Privacy and energy are primary concerns for sensor devices that offload compute to a potentially untrusted edge server or cloud. Homomorphic Encryption (HE) enables offload processing of encrypted data. HE offload processing retains data privacy, but is limited by the need for frequent communication between the client device and the offload server. Existing client-aided encrypted computing systems are optimized for performance on the offload server, failing to sufficiently address client costs, and precluding HE offload for low-resource (e.g., IoT) devices. We introduce Client-aided HE for Opaque Compute Offloading (CHOCO), a client-optimized system for encrypted offload processing. CHOCO introduces rotational redundancy, an algorithmic optimization to minimize computing and communication costs. We design Client-Aided HE for Opaque Compute Offloading Through Accelerated Cryptographic Operations (CHOCO-TACO), a comprehensive architectural accelerator for client-side cryptographic operations that eliminates most of their time and energy costs. Our evaluation shows that CHOCO makes client-aided HE offloading feasible for resource-constrained clients. Compared to existing encrypted computing solutions, CHOCO reduces communication cost by up to 2948x. With hardware support, client-side encryption/decryption is faster by 1094x and uses 648x less energy. In our end-to-end implementation of a large-scale DNN (VGG16), CHOCO uses 37% less energy than local (unencrypted) computation.