Is there any setting, or app, that will allow you to take pictures (or video) while keeping the screen turned off? Phone is a stock Droid Maxx.
If connected to an external display, you may be able to record, but you will need the touch screen to take still images. (Actually, all you need the display for, is centering your shots, and then using the button press to take the image)
Recording should be easy, just start the camera's record function and place the phone into lock, the recording should remain active, unless you have a small amount of external or internal memory to work with, and the video function shuts off automatically in XX seconds, but if you care saving to external memory, and have say, 32 GB free memory, you should be able to record for quite some time without restriction.
Look over your menu options, and settings, change your storage to external, and adjust the timing relationship for video recording length if possible.