Monitoring medical data, e.g., Electrocardiogram (ECG) signals, is a common application of Internet of Things (IoT) devices. Compression methods are often applied on the massive amounts of sensor data generated prior to sending it to the Cloud to reduce the storage and delivery costs. A lossy compression provides high compression gain (CG), but may reduce the performance of an ECG application (downstream task) due to information loss. Previous works on ECG monitoring focus either on optimizing the signal reconstruction or the task's performance. Instead, we advocate a self-adapting lossy compression solution that enables configuring a desired performance level on the downstream tasks while maintaining an optimized CG that reduces Cloud costs. We propose Dynamic-Deep, a task-aware compression geared for IoT-Cloud architectures. Our compressor is trained to optimize the CG while maintaining the performance requirement of the downstream tasks chosen out of a wide range. In deployment, the IoT edge device adapts the compression and sends an optimized representation for each data segment, accounting for the downstream task's desired performance without relying on feedback from the Cloud. We conduct an extensive evaluation of our approach on common ECG datasets using two popular ECG applications, which includes heart rate (HR) arrhythmia classification. We demonstrate that Dynamic-Deep can be configured to improve HR classification F1-score in a wide range of requirements. One of which is tuned to improve the F1-score by 3 and increases CG by up to 83% compared to the previous state-of-the-art (autoencoder-based) compressor. Analyzing Dynamic-Deep on the Google Cloud Platform, we observe a 97% reduction in cloud costs compared to a no compression solution.