Okay, nearly 3 months later, I have some additional information. During a bedroom remodel, we moved into one of the other rooms and used that TV with identical Roku stick and did not experience any battery life issues with that TV/Remove. A month later we moved back into our room and after those batteies quickly went dead again, I swapped the remotes with the other room. After two more month (plus the original usage of several months on that remote) the batteries finally went dead. We also used our original remote in the other room where they quickly continued to die. So, it appears rather conclusively that, in our particular case, the battery life can be directly related to the controller and not the stick.
What is the recommended path forward. I remote replacement?