I have the same problem, but with a Lifestyle 650 and the AppleTV 4K.
With Dolby Vision content, it's the worst. But no matter what content is played, it is not in sync.
Both of you should check how it's going when connecting a source device (PS4, ATV4 or whatever else) to a Bose HDMI Input. If there is no delay in this setup, the delay is caused by the TV when it's passing through audio from its HDMI inputs to the ARC port. In this case there won't be much that Bose can do about it.
What you can then try if you still want to hook your sources to the TV rather than the Bose is to change the audio output format of your TV from "Dolby Digital" or "Bitstream" or "Multichannel" (or whatever it might be called in a Samsung TV) to "Stereo" or "2-Channel" or "PCM".
My TV (Philips) adds an uncorrectable audio delay to all audio it sends out (no matter if coming from built-in sources like tuners or apps, or from external HDMI sources) when the audio format is Dolby Digital bitstream. The only way to solve that is to configure audio output to 2-channel stereo. That might help in your situation as well, however when doing so you will loose the Bose's capability to process 5.1 Dolby Digital sources. For simpler soundbars that can only do stereo this might not be much of a problem, but for the ST300, let alone LS650, this is quite a sacrifice. That's one of the reasons why I have opted for a HDMI switch that I'm using to hook up all my HDMI sources to the one HDMI input port of the ST300.
My AppleTv is connected to the lifestyle with HDMI and there are these problems with the sound. The sound comes first and then the picture.
Unfortunately, this only occurs with the AppleTV. My Oppo 203 does not have these problems.
If audio is before picture and not the other way round, you should be able to correct this using the audio delay setting in the Bose. Don't know about the Lifestyle systems, but the ST300 has such a setting.
Mar 15, 2018
Thanks for posting on the Community!
I would recommend BigBlue007 recommended regarding trying connecting the source directly to the soundbar either with the HDMI In or through the optical port. This will confirm where the lag is being introduced.
Regarding the AV Sync setting in the SoundTouch app, this was designed to delay the SoundTouch 300's audio stream so that it can sync with other SoundTouch speakers on the network when grouped together which will actually make the issue worse.
Tony A - Community Support