While working on a solution, I found electronic editions of the manuals for both your TV and your Bose system. These I find helpful for quick searching..
My initial answer is this..
If you are using HDMI, there is NO reason to ever hook up the RW Audio inputs on HDMI 2.
The audio inputs associated with the HDMI 2 connector are only for use if you have an old piece of equipment that uses DVI outputs. (DVI and HDMI are exactly the same type of digital system except for the SIZE of the connector and the fact that DVI does not carry audio). Since DVI does not carry audio, then you would need the RW audio inputs to carry audio instead. You do not have this situation, so do not plug them in...
Page 36 of your TV owners manual speaks of INPUT Labels..
It may be that you have somehow locked out the HDM input which is why it is not showing up as an option..
Terminals: Comp.1-2/HDMI 1-2/
Video1-3 • Label: SKIP*/VCR/DVD/CABLE/
* If "SKIP" is selected, the TV/VIDEO
button press will skip its input.
Work with this and see what you can get done.
Spend more effort into getting it set up so that you can SEE and hear each component before you worry about controlling everything with one remote or making the button labels match with what is actually happening.
Also, make sure that the HDMI cables and ports are working correctly..
Switch the ports that the cable box and the DVD player are hooked up with. See if you can heare and SEE the DVD player plugged into the other HDMI port.
If so, then you can switch them back and start with hooking it up right and reporgramming the TV.