This is definitely a first. I can tell you without question it does not apply to any Verts, nor have I heard from any Coupe owners that there is a problem- but again, I have no idea who many are that buy these and for what use.
Based on what you are saying, the problem is obviously a back-feed somewhere in the system. The Brake Lamps are being fed by the turn signal. It is likely how the system is grounded or the selection of parts you are using. I will take a look at the wiring diagram for a coupe and report back. Edit: Just looked. All grounds on the rear tail are common per side on the coupe, just like the Vert. This means that a few things could be causing this:
1. Poor chassis ground. All current at that small level should be going directly to ground, but for some reason it is back-feeding your brake lamps. Check grounds on each side.
2. Non-polarized brake lamp bulbs. {We don't use these for this reason} Polarized bulbs would be incapable of back-feeding like you are describing.
3. Load-resisted turn signals. {We don't use these for this reason} These will pass enough current to upset things, especially if mixed with non-load resisted brake lamps and non-polarized bulbs.
4. Any chance the coupe has "repeater" side markers like the front of the coupe/vert have from the factory? This incandescent bulb may be causing an issue as well. I honestly cannot recall if it does or not, and have all Verts here at the moment.
Also, just curious, what exactly did you alter or omit from the front turn signals in order to get them properly working? I assume the ground was altered or the outer bulb is being left out when adding the four primary LED's in the corners? Make sure that you have the "small" outer turn signal unplugged at this time to make sure it is not related to the back-feed, which it could be if it is still incandescent or bi-polar. This is highly unlikely to be the cause, but just figured its worth a quick "unplug" to isolate.