Checking trailer lights by yourself stinks. I like the idea of a wireless remote.
If you put a short battery in a longer battery box, that would leave space on the side and likely the top for mounting stuff. With the box longer than the battery, a portion of that space could be used to permanently mount the truck side connectors (3, 4, and 7 pin). Worst case is 7 pin with 6 signal wires. Use a 8 channel relay board to control the connection between the battery and the trailer. Since only one relay will fire at a time you may get away with a 5V relay board (I would roll the dice). Or you could find a 12V relay board and power it directly from the battery.