The cult claims all members retain their normal victory conditions, such townies are still townies, as the cults victory conditions (get rid of the mafia, etc) is the same as the towns. So Pizza invited to the cult would still retain his township, and thus, he would still be working in the interests of the town regardless of that.
Edit:
Major Robert Dump gave us more information on that here:
My goal, according to my cult PM, is to eliminate the mafia and keep the cult alive.
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