You could set your phone to stop turning off BT when it goes into power saving mode (I.e. when the screen turns off), although the Pixel 6 has reports of shutting things like wifi and cellular off when the screen turns off so it might not help.
But why do you want to use a (battery powered) phone attached to a (battery powered) eBoard to play a Daily game, where you might only make one move per day?
I just got mine today, and I'm really underwhelmed by it. It's been challenging maintaining a connection (Pixel 6 + Chess.com). If your phone's screen turns off, the connection usually fails too, and won't recover, and I have to quit the mobile app, totally turn off Bluetooth on the phone and turn it back on, then load the Chessnut app and reconnect.
Also, when I try to set up the pieces for a Chess.com game in progress and then open it in the Chessnut app, even though I have a green chessboard icon (showing the board is sync'd), it's not really sync'd and so moves aren't registering.
I decided to try the Chessnut app on my MacBook Air, and at least this time I see a more detailed error:
Is it not possible to play Daily games with Chessnut? I did make a couple of moves earlier and it worked fine, but now it's not.
I don't get the hype. This thing is super buggy and not ready for prime time.