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

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.

Mr. Chesky's -6dB minimally controlled sub-bass speakers are extremely forgiving of room placement simply because they don't output frequency low enough to excite a rooms bass modes.

A properly positioned and optimized -3dB subwoofer can provide the delicate subsonic presentation for almost half the price. Some offer remote dsp control and a few customizable presets to compensate for Cheskey's or your own personal taste. Way more fun.