BC> I wonder if I can build up the height of the affected area somehow,
 BC> with the largest buildup in the centre and then sloping away towards
 BC> the sides so that the water will drain out to the street on one side
 BC> or my driveway on the other.  Or will that just move the problem
 BC> further 'up' the lawn?
Or further back towards the house, which is the LAST thing you want.
You might try a slight swale leading the water to where it can run away.
 BC> How would you build the lawn up?  Sand?  Would you have to re-seed, or
 BC> would it grow through? 
If you don't have to do so much that you bury the crowns.
... Would a homeless Bugs Bunny work for Fudd?
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