It is very unlikely that the OBDII power cable along with the logger could drain your battery.
The TripLogik TL1000 uses between 40ma and 50ma when it is activated and is looking for satellites.
When stationary and has a fix on the satellites, it will turn off the LED and reduce the current consumption to 35-40ma
At peek use, it could be between 50ma and 55ma with the LED on full brightness
(40ma = milliamps or 0.04 amps)
Car batteries are usually in the range of 60 to 120Amp/h ( trucks and pickups are usually bigger)
Depending on the Amp per hours (Amp/h) of your battery, you can calculate how long it will take to drain your battery with the following link.
Assuming your TripLogik TL1000 logger and the power supply cable uses an average of 0.050 amps ( 50 miliamps ) under 12Volts
We can calculate
12Volts x 0.050 amp = 0.6 watts
According to the calculation, If your car has the smallest 60Amp/h battery, you should be able to run your TripLogik for 500 hours (almost 21 days)
The average car has a 120Amp/h batteries so it would be closer to 1000 hours or 42 days.
Of course, if you don't intend on using your car for over a month, it is a good idea to remove the TripLogik and any other electronic accessories from the car.
By definition, typical TripLogik users are mostly on the road, so battery discharge worries are never an issue.
Mechanics are sometimes quick to point out anything they don't know, be aware they will suspect it immediately.
Ground and electrical fault are a common problem in cars, they are usually elusive to find.
If the mechanics makes is mind on it, he may never look anywhere else.
More info about batteries