grammatical number - Plural of Friday 13th?

I would argue that "Friday the 13th" (with the the) is such a common set phrase that it behaves as a syntactic atom, i.e. something that shouldn't be split up by syntactic processes. In my experience, people almost never pause at any point when pronouncing the phrase "Friday the 13th", and it all has one continuous intonational contour.

This is all to back up my intuition that, as a native speaker (of American English), Friday the 13ths is the only plural form I can possibly imagine myself speaking here. It does look a little funny in writing, but so does Fridays the 13th, so I'd go with the more natural spoken form.