LJBF is a feminine social contrivance. Women have used the LJBF rejection for a hundred years because it serves an ego preservation function for her. To a greater or lesser degree, women require attention and the more they have of it the more affirmation they experience, both personally and socially. The LJBF rejection has classically ensured that a woman can reject a man yet still maintain his previous attention. It also puts the responsibility for the rejection back on his shoulders since, should he decline the 'offer of friendship', he is then responsible for entertaining this friendship. This of course has the potential to backfire on women these days since the standard AFC will accept a LJBF rejection in the mistaken hopes of 'proving' himself worthy of her intimacy by being the perfect 'surrogate boyfriend' - fulfilling all her attention and loyalty prerequisites with no expectation of reciprocating her own intimacy.
The LJBF rejection also serves as an ego preservation for her in that having offered the false olive branch of 'friendship' to him in her rejection she also can sleep that night knowing that she (and any of her peers) wont think any less of herself. After all, she offered to be friends, right? She is absolved of any feelings of personal guilt or any responsibilities for his feelings if she still wants to remain amiable with him.
I think the default response should be to take a LJBF as a rejection (and her loss) ergo, you remove the reinforcer - attention. Up until the point you made an approach for her intimacy she was enjoying the benefit of your attentions. After a LJBF response her latent intent is to keep that reinforcer of attention. Do not reward her for this disingenuous response, she will only use it on you again or with another guy in a similar situation since it was reinforced the last time this circumstance was experienced. And should the next fellow reinforce it further she will internalize this as her standard response.
Obviously the best way to enact this is to use a takeaway and turn down her LJBF. An outright refusal of her pseudo-friendship offer would be ideal, but not always possible given social settings, however a takeaway is always warranted. Mmost women, whether serious or not in their LJBF rejection, will almost always follow up with some kind of communication when you do remove your previous attentions. This was a previous reinforcer to her and like most animals when faced with a behavioral extinction, she will attempt to re-establish that reward. This is why if you do end up cutting all contact with her she will have a tendency to pursue - and depending on the individual sometimes more actively pursue - your attention, even if she has no intent of becoming intimate with you. Women do this in an effort to maintain self-affirmation (i.e. she wants to verify everything is 'OK' between you and her in an email or IM) after a rejection.