"And Lucy perhaps at first might think only of procuring his good offices in my favour. Other designs might afterwards arise."

Is this a matter of benefit of the doubt? Excessive charity? (-; With all we've come to know of Lucy, does anyone actually think other designs arose afterwards for her?

"perhaps at first accidentally meeting"

In the infamous words of Alan Rickman, "I think not."

My theory on this is that Lucy was looking for a weak link in the family, so purposely set out to 'accidentally' meet Robert. Her foremost design was to get the money she had been counting since before her engagement. Since that money now belonged to Robert, and she undoubtedly could tell he was an intellectual giant (-; she could set out manipulating him. One possibility for getting to him was brotherly aid. She had to maintain the facade of wishing to keep the engagement because unless Robert believed she still desired the marriage, she couldn't use the pretense of looking out for Edward's interests to get close to him.