Nice work on the boxes. They look great.
I suspect your problem is the 9 volt adapter and the 9 volt battery do not supply enough power. Check the current rating of the adapter. I hope you have the adapter power and the 9 volt battery going into the external power connector on the arduino. Are you powering the external devices off the Ardiuno internal 5 volt pins?
Perhaps a block diagram showing the pieces and the power and ground connections would help.
The heaviest load the Nano has to carry is (I think) for the transistor that switches the remote led on/off. It should be no problem, but I'll try running the program without switching it on. Who knows.
My problem is this: the whole thing works great, on USB power. Yet when I connect the 9v adapter, it works for a couple of seconds and then crashes -- always while updating the TM1637 at the far end of the cable.