A very interesting post. I never dug deep enough into the RFC to see why this might be a problem. Our servers deal with a lot of small transactions (Usually only one with no additional connections) so it hasn't been a problem.
For a quick fix, what do you think about a module that would use IO::Scalar or Tie::Handle or something similar to redirect STDOUT to a scalar and output the Content-Length followed by the scalar (presumably containing HTML) on CLOSE?