If placing a sub in the middle of thee room next to your couch, you'd want to go with as big and powerful one as possible, since it will have zero room reinforcement. So I'd recommend starting with AT MINIMUM a Hsu VTF-3 or VTF-15 ... you can easily throw some sort of fabric over the top plus a lamp or plant to disguise it, to make your wife happy.My issue with sub placement in this room is that I don't really have any corners at all. There is a hallway to a bedroom at the corners on either side of the TV, and behind the listening area is wide open and includes the kitchen and dining room. Directly under the TV is going to be a floating cabinet with relatively low clearance underneath. My only option for putting a sub against the wall would be some DIY enclosure that was very wide and short to slide underneath it, which would ruin the look of the floating cabinet. I can't imagine it would be possible to place a sub underneath the left and right speakers, at least in this application where they're not behind an AT screen.
And I'd try to economize on the front L/R and surrounds. I'd look at the Hsu HC-1 for the center and HB-1 for front L/R and their HIW-1 inwalls for surrounds. These are all reasonably priced, aesthetically acceptable, have decently high sensitivity for your Denon X1200 to easily power, and will fit easily into your budget. More importantly, they are slightly rolled off in the treble unlike most horn tweeters, which will insure no ear-bleeding brightness given all the hard reflective surfaces in your space.