Problem solved.
I hacked together a DW extension to do this at the touch of a function key.
Most of my markup files are HTML fragments, so the extension prepends the minimal necessary header items --including the appropriate DOCTYPE-- to the file begining, postpends the necessary closing tags at the end of the file, then runs the normal CS6 W3C validation. Then the extension removes the prepended and postpended items. It shouldn't be difficult to add a test so the additions (and subsequent subtractions) are only done on fragments, but I haven't bothered. The DOCTYPE is fixed, HTML 4.01 Transitional; a more clever extension would remove that limitation, too, but I'm not clever enough.
In several weeks of use, this has worked without any evident problem. It is fast, too!
For users who are constrained by security considerations from submitting work to the W3C for validation, I understand it is possible to install a local version of the W3C validator -- with some technical skill and some luck, as I've read -- it isn't trivial. For tekkies without a 24/7 net connection, this might be a reasonable alternative, too.
For less technical folks in that situation, it might still be useful to have access to Don Booth's "old, limited" validator on CS6. No systematic results, but is my impression that the old validator and the W3C are approximately equivalent for the kind of work I typically do.