Would it be possible to cut some tile to fill in the 3 inch gap? (leaving room for grout lines)Then put down mortar at an angle to give the 1/2 inch rise, and then push the tile into the mortar to get the right angle for the transition.
Or take pieces of tapered shim stock or cedar shingle to raise the 1/2 inch over the 3 inch distance. Fasten down firmly with PL400 and nails. then put the mortar and tile down. Since that area doesn't get that much traffic or loading, there shouldn't be any problem with tile integrity.
Or, how about a piece of oak trim cut to the right size with the correct taper. Sanded, stained and varnished, it could look good. If a door goes in the closet doorway, it will not be that noticeable.