The use of privately gathered evidence intended to be handed over to law enforcement may violate the Fourth Amendment via the exclusionary rule.
That insurance customers are contractually obligated to assist the insurer with investigations may violate the Fifth if that contract is being abused in this way, as well.
Yes, when third parties start acting under the direction of the police, they become the police in the eyes of the constitution, with the same restrictions.
That has nothing to do with the GGP’s strange claim that there’s some general constitutional prohibition against laws with “an inherent conflict of interest”.
That insurance customers are contractually obligated to assist the insurer with investigations may violate the Fifth if that contract is being abused in this way, as well.
https://en.wikipedia.org/wiki/Exclusionary_rule