I can't come up with anything solid as to why Lt. Bronaugh was with Company "D" at Pine Bluff. But here are some speculations.
3rd Lt. was enlisted at Pine Bluff and assigned as the company Adjuntant (Book Keeper).
Donalson MacGregor, Captain of Company "D", was a very influencial and power planter in the Pine Bluff Area and friends with and even related to ex-arkansas Governor John S. Roane. Donalson was also related to ex-President of the United States Andrew Jackson.
So while as you speculate that there may have been a connection through the Masonic Lodge, there was more than enough political connections to go around to indicate that these mens paths crossed each other regularly.