Ron -
I like your idea for tapping in to the TT's running lights for power to the booster (I too always tow with the lights on so would only be hot while towing).
Where the issue in your plan could be is that per Minder Research/Valterra Products you want the weather-proof booster's antenna outside the TT such as mounted to the frame, inside the mounted spare tire, etc...
However, I don't think that is truly a serious issue. Since you have the access area and you know you want to splice in to that line you could just set the booster inside the access area and give it a try. If there is trouble you can always drill a small hole through the floor and mount the booster under the trailer.
I really think you might do a test tow or two without the booster as even your 1685 is a pretty short trailer. You might find that it's not necessary.
By the way, 1575's have that same false wall/access area. That is where I mounted my Victron battery monitor. If I do decide to use the booster I will definitely follow your lead and use the marker/running lights wire. Thanks for the idea. 
*From here -
https://community.lanceowners.org/top...6#415772502841291706