Most electrical, energy and fire codes require fluorescent or high efficiency lumens in closets. Color corrected bulbs or warm toned LEDs will mitigate the harshness of the fluorescent. A single 4' fixture n the center, behind the wall will limit the wiring some. The two doors and the three way switches are more of an issue. SIPS are tough to wire and it is expensive. We have recently begun using wireless switches in similar situations (like on a pocket door wall). Available from Lutron, they require only a radio receiver in the ceiling to function. They can be mounted in a false box and cover plate to keep the look consistent.

There are battery operated lights that could be attached to the ceiling without having to wire them.

Thanks Ironwood. The Lutron LEDs look like a good possiblity. Don't know how I feel about a bigger radio wave bath but anyone with SIPS should appreciate this option.