The Garmin eLog ELD adapter should remain plugged in at all times. This will ensure you remain FMCSA compliant and avoid a power compliance malfunction.

Create a RODS Report before getting it serviced to ensure you have your driving data available in case of an audit. A RODS Report cannot be created, saved, or exported without being actively connected to the Garmin eLog.

Drivers: How do I Submit Records of Duty Status to my Motor Carrier from a Garmin eLog ELD? (Exporting a RODS Report)
Motor Carrier: As a Motor Carrier, How do I Export or Submit Logs from the Garmin eLog App for my Drivers?

Log out of the Garmin eLog App while connected to the Garmin eLog ELD adapter before the semi-truck is serviced. Logging out after disconnecting from the Garmin eLog ELD adapter can cause the driving time to be recorded against your profile.

All driving time done by a mechanic would be recorded to the unassigned driver. The Motor Carrier could then note those unidentified driving hours indicating it was driven by a mechanic while in service.