Power Saving Mode (PSM), a power-saving mechanism in LTE, was specifically designed to address such issues. It allows the device to inform the eNB (base station) that it will be offline for a certain period while ensuring it periodically wakes up to perform a Tracking Area Update (TAU), preventing the loss of registration. This concept is similar to Session Tickets or Session IDs in (D)TLS—or at least, that’s how I like to think about it. However, there are no guarantees that the operator will support this feature or that they will support the report-in period that you want!
Maintaining an active session for communication between the endpoint and the edge device is highly power-intensive. Even with (e)DRX, the average power consumption remains significantly higher than in sleep mode. Moreover, the vast majority of devices do not need to frequently ping a management server, as configuration and firmware updates are typically rare in most IoT deployments.