I have the exact same problem. As soon as my phone locks or even if I just put the Roku app in the background it deactivates the private listening mode. Only way I can get it to stay on is by having my phone plugged in, my phone unlocked, and the app open and actively running.
I too have made sure the app has permission to run in the background, and have turned off battery optimization. I am using a #Roku Ultra (4800 series) My phone is a Samsung Galaxy s20FE.