Even if someone wanted to blame the Holy Church for this, they couldn't. The Minister of War did not use the Holy Essence.
He also did not serve the Holy Church, but rather worked for the Holy Emperor. Everything was carefully arranged to implicate the Holy Emperor.
If there was another envoy, they may have fallen for this trap, potentially even executing the Holy Emperor.
Unlike Eren, the other Great Elders were more ruthless and uncaring. If someone attempted to kill them, they would not hesitate to eliminate that person and their entire family.
If there was truly an Envoy from the Great Demon Sect, they would not have cared that they were in the Western Empire or within the reach of the Western Overlord.