Plea for help:
I'm sure the electrical engineers will have better ideas. That said, you mentioned powering the Raspberry off of 3.3v if I heard you correctly in the video. I'm assuming that was some confusion on my part when listening and I wanted to confirm you are actually powering it off of 5v from the cape.
Yeah the 5v is powering the Pi but the 3.3v connector (which may be unnecessary seeing as it isn't on the website) is connecting the 3.3v on the Pi to the 3.3v on the cape. Sorry about the confusion.
Can you verify that the Pi has 3.3V, 2.5V and 1.8V on the Pi board? How accurate is the 3.3V on the Pi board? If you are sending 5V from the cape to the Pi what is the exact 5V voltage as seen by the Pi.
I think the LED you see on for about 1 second is on the Pi 3.3V bus with no software control. If that light goes out I would guess you stop getting power from the ROV cape.
Are you sure that your BeagleBone can provide enough current from its pins? Looking it up online, it looks like it is roughly 8mA per pin, while the RPi can take up to 700mA. Also, your RPi needs only a 5V input, IIRC. When I've powered one before from a battery, I used a 5V BEC (like this one: http://www.hobbyking.com/hobbyking/store/__15212__hobbyking_micro_u...) to drop 12V down to 5V. I then cut up a micro USB cable to power the RPi from its USB port. Hope this helps!