Hi Janey,
Towing our Jeep Liberty, we are setup the same as Tom Hanlon with regards to power to the tow vehicle.
We have a strange thing going on with the electronics with our tow setup that I can't explain. Everything works as it should with one exception which might also be happening to you.
When we shut off the PC, the back-up camera we use as our rear view mirror, is still on, along with the tiny lights under the PC's front doors. When traveling multiple days and we stop for the night, I unhook the umbilical cord to the Jeep and all is 100% shut down. There is a diode in the Jeep to prevent back-voltage to the PC, but it's still happening, or something else is happening that appears to be back voltage. The power being supplied to the camera & lights might be a PC issue or not. I never researched why it's happening. At the end of the day, still hooked up for the night, I simply unplug that cord to shut down the camera & tiny outside lighting.
I shared this because just maybe the same thing is happening to you. The two vehicles hooked up together overnight, your CR-V battery may be draining.