The inner sun will then reflect it's light waves off of the transparent perspex
LOL. Do you know what the words "reflect" and "transparent" mean?
You do get partial reflections off transparent media like water, glass, and plastic. Lens makers go to great lengths to minimize reflections, but they can't be eliminated entirely.
To sceptimatic, I respectfully suggest you not try to evacuate the space between your inner and outer domes - at least not at first. Any loss in fidelity will be minute - in fact, it might be better if you just painted the outside (convex side) of your inner dome flat black so you don't get a second reflection off that surface. Evacuating the space behind it (if even possible, which is doubtful) will make those reflections (slightly) worse.
Getting a structure of that size made of those materials to hold even a partial vacuum will be a difficult task. To keep it from collapsing if it could be evacuated would be monumental.
Do you realize how much force your domes must withstand? A disk 12 feet in diameter has an area of 113 square feet, or 16,286 square inches. If you reduce the pressure between two disks this size by as little as 5 psi, that's 81,430 pounds of force pushing each side toward the other. That's the best case. If they're domes, the areas (and, thus the force) will be even greater; a hemisphere 12 ft in diameter has an area twice this. Never mind, you don't believe in math. Carry on! Do please keep us apprised of the progress, and don't spend too much valuable time trying to evacuate the space between the domes - I await your results.