My guess is that Jane is fully aware of this situation, since she's an HR executive at Company A. I'd think that a company would stop pushing after a candidate rejects their 3rd job offer. The fact that Company A continues to reach out after repeated rejections makes me think there's a non-zero chance Jane is involved in it. Bullies don't change; LW is right not to work there, because Jane is in the perfect position to harass LW.

I think LW should reply to Company A one final time, firmly but politely state their answer is final, thank them for their time and interest, and leave it at that. Block them if they continue after that, and maybe talk to an employment lawyer in case other weird things start to happen.