This is a 2015 Jayco Jay Flight 26BHS...? I tried to look at the features and it shows "Entertainment Center" as a bookcase-looking item between Living Room and Foot of Queen Bed. You say that swivels? Confusing! Please confirm the Year/Make/Series and if you can post or point us to photos, please do. Is that Mirror made of Glass or a reflective plastic? Could you re-mount the mirror with moldings around it?
I know you're trying hard to describe this thing, but all I get out of it right now is the location is essentially a particle-board bookcase with a thin back that has a mirror stuck to it. If that's what it is, I'd think of installing a stiffer back, like Westend describes, secured to the inside of the sides with brackets or angle aluminum. Put Tee Nuts on the backside of the new back, to secure the TV bracket to without having to pull the mirror and still rely on thin luan for the bracket.
Still, I wonder if I'm on track at all. Pictures would really help. Remember, they have to be on a website someplace, then linked to RV.net using Advanced Post Form then the yellow postcard icon.