Actually its quite easy to overthrow the purists, look at the policies,
get rid of narrato veterans,
get the political advisor that boosts popularity in intellectuals,
and change other policies that make the purists more popular
and bend them in your favour.(costes political power of course)
and try to get rid of the enclave scienctists and officers with the focus tree(if it requires stability then try to boost it with shortening admin shifts)

Then just let their popularity go down slowly or boost it with a speech or a couple of murders and you'll overthrow them in no time.

This is coming from someone who found out only before it was too late, and after 2nd attempt, so it should be easier for you to beat them now