Though it might be nice to better isolate the trailer electrical from the bike's electrical, when I did my GoldWing trailer I didn't bother with the converter box to get the separate turn and brake light signals, I just ran a second small one-wire plug alongside the standard 4-wire plug. It worked for me because I had added extra lights on the rear of the trailer box, so the lights I used as brake lights on the trailer could also be separate from the turn signals.
I think that if you only have the one pair of lights on the trailer and they have to be running, brake, and turn signal, then you will need the converter box.
In addition to the stock light assemblies on the trailer frame that came with 1157 dual element bulbs, I also added a pair of 4" round light assemblies with dual element bulbs to the rear of the trailer box. I had it wired so that the lights on the box were running and brake lights, while the stock units on the frame were brake lights and turn signals. People behind me could really tell when I hit the brakes!
With the trailer connected the turn signals flashed a little faster because of the higher load, but it wasn't bad. A guy could always pop in a heavy duty flasher to slow them back down if he felt it needed it.
All of the wiring on the bike that I needed to tie into was under the seat, and when the trailer wasn't connected the plugs on the bike could be stuffed up inside the license plate holder. Everything was all bundled and taped up nice. I ran it this way for years and it was fine.
I think your CX will have adequate juice to run the lights, its not like your brake and turn lights are on all the time