What tool do I need for measuring DC power consumption of device over a period of time?
There are few options for such a task:


A DMM with data logging capability. You need two of them, one for current and one for voltage.
A DAQ board, an analog front end to convert the current into voltage and a PC. You will need one with 2 channels and sufficient resolution and for your requirements.
A purpose build device for such task - usually they are called IoT power monitors or something similar. Such a device implements the ADC and analog circuitry, you will need a PC to complete the system.