Yea great idea, but a small problem here, and that is said design, whilst it may be fun to think up an idea it is another to design it. A couple of thoughts:
A Fresnel lens on the window to focus microwaves onto a small aerial in the room.
A see though dish made from fine spluttered metal that is almost a mirror.
A communal dish on the roof and out of site, probably the most practical, but that requires agreement. difficult one that, but certainly the most practicable.
Not watching satellite at all, stick to Hi-Fi, no dish, no problem.
But really that flat panel aerial does look good, and better than most.
John...