I believe you can store it under there. Without wiring a switch, the BT OBD adapter will draw power though.
Here is the one recommended by the TM Spy guy:
LELink Bluetooth Low Energy BLE OBD-II OBD2 Car Diagnostic Tool For iPhone/iPod/iPad https://www.amazon.com/dp/B00QJRYMFC?ref=yo_pop_ma_swf