What? Can you put in probabilistic terms what "this is not the case" is?

There are an infinite number of models where p(HHTHT | model) != 1, or where p(HHTHT | model) = 0. We need to know which one you're referring to, in order to calculate a p-value.

I think you have made a serious error by believing you can simply "reverse" the model p(HHTHT | conspiracy model) = 1, p(everything else | conspiracy model) = 0.

If the null hypothesis is a fair flip, then the alternative can't be a conspiracy, because the null and alternative need to be complementary statements. So if the null is fair flip, then the alternative is "not fair flip".

edit: whoops, changed mutually exclusive to complementary. see http://www.its.caltech.edu/~mshum/stats/lect8.pdf