Curiously are you using an optical cable or HDMI cable to connect the SoundTouch 300 to the TV?
The first step to configure the Bose remote to power on both the SoundTouch 300 and the TV at the same time would be to program the TV into the Bose remote. We have the manual on how to program this linked below:
After you have confirmed that the Bose remote is able to control the TV, on the SoundTouch 300 remote control, simultaneously press and hold the TV source button along with the appropriate source button for 10 seconds until both buttons flash three times.
Let us know if that helps resolve the issue you are experiencing! Thanks!