You can use a small MOSFET to switch the current. I would use an N-Channel MOSFET, connect the Drain to the cathode (negative side) of the LED and the Source to the negative battery terminal. The anode (positive side) of the LED goes to the positive battery terminal. The output of your motion sensor goes to the Gate of the MOSFET.
But again - it must be a true "logic-level" MOSFET which will definitely switch at 3.3 V.
The popular (and cheap) 2N7000 or more correctly the 2N7002 version may work or if more (than 100 mA as I suspect it is) current is required, perhaps an AOD4184A.