You could connect a speaker to line level adapter instead of the internal TV speakers and connect a BT transmitter to it.
You could also use an HDMI input selector with audio extraction to give you an audio output for a BT transmitter for those sources. If the soundbar has an aux input you can connect it directly without bluetooth.
That's difficult if there is no audio out. If you are playing from a PC, you could use the soundbar via BT, and set that as the primary audio device instead of the HDMI output, but then it would only work if it's a PC.
You could connect a speaker to line level adapter instead of the internal TV speakers and connect a BT transmitter to it.
You could also use an HDMI input selector with audio extraction to give you an audio output for a BT transmitter for those sources. If the soundbar has an aux input you can connect it directly without bluetooth.