I found some time this week to finish assembling the platforms. After cutting all the parts carefully to size, I screwed them together with 3" screws.
I also cut the tops for the platforms to size. Then I took the bases up and spent considerable time getting them not only level, but level to each other. That will result in an even reveal when the baseboards are in place.
It took be the better part of an hour to level them properly. The floor slopes slightly from the north side (far wall in the picture below) to the south. It also slopes downward slightly from east to west (right to left). As a result, the platform base is resting on the floor in the far corner, but raised about 1/4" at the near corner.
I then leveled the platform for the bookcases to the same height as the window seat platform.
Before I could fasten the platforms down, I had to feed the electrical wires into the base platform and connect them to each other temporarily, so the room’s circuit would be whole. Then I fastened the platform bases down with Kreg pocket hole screws. While these are often used to fasten face frames together, I’ve found them excellent for this purpose. The last steps were to screw down the platform tops to the bases, cut and break off the excess shim bits, and clean up.
Now that I have level, sturdy platforms, I can install the tall room baseboard along the south wall and paint it. Once that is done, I can finally install the baseboards for the west wall.