Have not found a solution to this. However, this is also a bad solution because the sound via Bluetooth is usually out of sync. The Toslink output with an ADC is the better solution, also in terms of sound quality. I use: Oehlbach DA CONVERTER
Oehlbach DA CONVERTER - 192kHz / 24bit Audio Support (D/A Converter)
Depending on the stereo system
TV cinch connection (red/white) = stereo signal to stereo system. Always works, proven technology
TV HDMI 2 connection (ARC, Audio Return Channel) = Dolby, DTS, PCM, ... to stereo system, plus a few additional options (depending on the stereo system or AV reveiver) Car on/off, volume TV remote control goes to stereo system
SPDIF (digital audio signal) connection = Dolby, DTS, PCM, ... to stereo system.
I have compared many devices in this size and price range and read test reports. Now the device has been running for one and a half weeks and I am absolutely thrilled. The picture quality meets my expectations the most, even dark areas are still clear and the luminosity is great. Installation is easy. The sound runs through the hi-fi system with the help of a DAC. The device is really recommendable. Unfortunately, as a non-gamer, I can't answer your specific question. But take a look at the test reports on the web.