There are, of course, going to be boundary cases that don't work as expected as soon as you start playing with allowing undoubled double-quotes inside of a format that expects them doubled. However Text::xSV allows you to define arbitrary filters that it preprocesses text with, and should do a reasonable job on the above with the following filter:

  sub {
    my $line = shift;
    $line =~ s/\r$//;
    $line =~ s/"(.)/""$1/g;
    $line =~ s/"?,"?/,/g;
    return $line;
  }
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Yes, there is some fragility, but it should be at least moderately hard to trigger.