Has Lady Bercilak succeeded or failed in achieving her goal? What was the goal?

Yes, the Lady succeeds in her seducation. The most obvious symbol in Fitt III is the green girdle which Gawain secretly accepts from the lady. As discussed above, it is a deceptive object, for it claims to protect a man, but in this case has only caused Gawain to breach his moral code and (as we will see) ruin his sense of self. Although Gawain accepts it because of his fear of death, there are still all the trappings of romantic love: the lady unties it from her waist and wraps it around Gawain's. On the outside, it still appears as a love-token, thereby emphasizing the sense of deception when Gawain hides it from the lord. Also, of course, it is green, linking it immediately with the Green Knight whom Gawain must meet the next day. In a sense, it is a sort of a reverse-magic to that of the supernatural, indestructible knight ­ or at least Gawain hopes so. Yet both the Green Knight and the green girdle seem to hark from a world of the magical, the otherworldly, the natural and fertile and indestructible. Again, there are pagan connotations with the obvious emphasis on fertility. We can even see the pagan, magical green girdle as representing everything that is not acceptable by chivalric and Christian standards: in keeping it, Gawain goes against his code of honesty, courage, and faith.