11' is not a standard, all studs would have to be field cut, plywood only 8,9 & 10' lengths, drywall is 4' & wide board 4' 6". So you have several factors which will drive the price up on the build of the home by going with 11' ceilings. You can get a 10' pre cut for framing and 10' exterior sheeting for walls, drywall will still require a filler however 10' is a better option than 11'. Is this a CBS block exterior walls?

You can opt for the frame work of solid wood or waterproof plywood , polish it , can install lights at cross section , I don't know about the size and shape of your room . This way you can have ceiling fan also at the location you want . Easy and simple solution. will look elegant as well

Place the cabinet against a different wall and not beneath that beam? There's no need for a false ceiling, instead you could do a simple cornice to soften and enliven the hard line where walls meet ceiling. You could apply the same cornice to the short, hard line where the lower side of the beam meets either wall. Keep it simple. You could even apply cornice ONLY to where the lower edge of the beam meets the walls. It softens the heavy horizontal impression, allowing it to pretend to be a very slight arch. If you're going to repaint, consider using slightly different shades for wall and ceiling. An alternate to the cornice is a shallow horizontal ledge to hold LED cove uplighting along all the walls, at the same height as the lower edge of the beam. If you do this in wood, clad the lower surface of the beam to make a feature of it and to match the cove lighting ledges. Keep the design simple. If you do none of these things, explore Houzz for pleasing and simple lighting solutions to brighten up the room. Just a few ideas...

I will suggest you to have a wood look ceiling with black ceiling mounted lights and a vintage fan in the center. The flooring can be heritage/modern Bharat flooring series. The Arches can flank on two sides with Moroccan pots and planters. Let us know if you need any help. Thanks