I feel that we may be ignoring the psychological aspects of having a strong BATNA here. It's easy to say that you should just "ask for a better offer", but it is waay easier to actually do that when you know how much you are worth to another employer.
Some people may be able to pull off something like what you describe, but all research I've read on negotiating indicate that knowing your BATNA and having a strong one is key to achieving good negotiation results.
Yes, a big part of negotiation is knowing what you should do and how you should react despite your internal feelings about your situation. I know I've looked utterly disgusted at offers that were already making me giddy on the inside.
It's always nice to have multiple offers to choose from, but sometimes you're just not in that situation. When that happens you can still master your own psychology, keep your feelings to yourself, and negotiate with confidence.
Some people may be able to pull off something like what you describe, but all research I've read on negotiating indicate that knowing your BATNA and having a strong one is key to achieving good negotiation results.