Subwoofer - is there one that would work in this situation?

single sub can work, but not if you don't have ability to place it anywhere in room...in recent interview David Chesky uses REL T/7's with modified Quads...

That's what I suspected.

Thanks

Everyone is gonna say you need two.  Two are better.  But one is better than zero if you can adjust the roll off and volume to 'barely there'.  I had one single sub exactly where you describe in a Salamander three wide cabinet under the TV.  Even had a door with acoustic panel in front of it and it worked fine.  Good luck!

I think you will struggle with a single small sub in a locked location. DSP for the sub through internal or external like a Mini DSP would really help. 
 

if you plan to just come in under your mains, say 40hz or less, one can be ok. If you want to do a proper highpass at 60hz or higher two is much better/needed. 

Boxing in a sub sounds is a recipe for disaster. Adding vibration to your other equipment in the same shelf? No No Nanette!!

Buy the smaller REL T5x I use. I can be hidden many places.