I could never underatand why Lynn slept with Ed Exley in L.A. Confidential if she had feelings for Bud and she also knew how much he hated Ed.
She did it because Patchett (her employer) told her to.
She was, after all, a high class prostitute. Moreover, various conspirators wanted to get some dirt on Exley do derail his investigation into the Nite (or was it Night?) Owl killings. The photos taken during their tryst were clearly used later to enrage White and encourage him to try to kill Exley. And Patchett had form for this sort of blackmail having clearly used it to influence politicians and the DA previously.
The complicating factor is that it wasn't ultimately Patchett who used the photos, but Smith (who may have taken them directly from the photographer and gossip columnist Hudgens). Patchett may have wanted them for a more subtle use, but Smith exploited them for a more immediate end as he needed to neutralise Exley quickly and already had a direct link to Hudgens.
Lyn Bracken was just doing her normal job, though she clearly regretted the harm she caused to White. She was also clearly under duress at the time so she didn't have much of a choice to refuse a "normal" role just because of her feeling for White.