Check whether your TV saves audio settings per input. Many sets do, which is perfect here. Go to the console HDMI input, enable Game Mode there, set audio to Passthrough on that input, and set lip sync to zero. Leave the streaming app inputs on Auto if they already work fine so you do not break what is already in sync.
On the bar pick a neutral mode and disable processing. If the bar still trails, add an extra tick of video delay for the console input only. That way streaming stays untouched and your console gets its own tuned sync.
