Okay, so I learn something new every day on this Forum. I have had my Mopar harness for over a year and never noticed what you are talking about. I just hooked up to check it and sure enough I see the same thing you brought up. With the turn signal on, the Jeep brake light overrides the turn signal. I did notice that there is a slight variation in the brightness as the turn signal cycles. Like you, now I have to decide if I want to install some kind of switch or relay to kill the Jeep brake lights. The monitor on my RVI2 brake system lets me see when it is braking, so I guess if I'm aware that this happens I can just make sure to "pump" the brakes so I don't get a continuous outage of the turn signal. Another possibility would be to pull the brake light fuse when towing, but that could be a PITA!
It makes sense though, since the Mopar harness uses small relays to activate the turn/brake signals from the motor home. When the relay cycles off for a turn signal, it's still getting power from the Jeep brake lights. Others with the extra bulb installed in the tail assembly have reported the same thing, both operating at once.
Dan & Ro with two Rescued Boston "Terrorists"
2015 Tiffin Allegro RED 38QRA
Previously '06 Itasca Sunrise 35A on a W22 Workhorse
'12 Jeep Wrangler Sport and/or '08 Honda Goldwing