[@ProPro] My initial thought was that the White Lotus had to change its play style significantly after the Red Lotus played their hand. In Aangs time, the White Lotus was comprised of a few elite masters. They diluted that in Kora's time and became more like glorified body guards. So when the Red Lotus, a group of a few elite masters, showed up, the White was utterly wiped. My thoughts were that when the Red were forced underground, the White focused its efforts on hunting them down and slipping back into the shadows after them.

Fan wasn't employed by the White Lotus. He was TRAINED by them, and works for them as an operative, responsible for stopping threats to the avatar (like Kora's kidnapping) before they happen. Keeping the Red Lotus scattered and trying to keep them from unifying again.

The idea is that Fan sueded in a mission but was injured to an extent that he could no longer work as an assassin for them, but his knowledge was still useful. Something to prepare the Avatar for the manipulations of the world around her