If they knew what was about to happen, Thanatos wouldn't have volunteered for it. Not only because it would certainly be easier for him to wrestle Orpheus than for him to be wrestled anywhere that he doesn't want to be, but because frankly, that is just unbecoming of a man of his position!

"Hear me, shade", Thanatos starts in the most commanding voice that he can manage, his face still as stony as ever when the thing shrieks at him. In annoyance, he stamps the bottom of his scythe against the ground. "On my authority as Death Incarnate, I grant you leave of this place."

He doesn't really have the time to react when the thing rushes him, though in truth Thanatos believed it was simply as incorporeal as a shade(which it was) and was intending to go past him(which it was not). It is perhaps a mercy that when the icy fog envelops him and finds it's way into his body, Thanatos drops his scythe as he is possessed. He's left grasping at his head in some confusion in the seconds between his possession and losing his ability to control himself.

With only a short struggle, Thanatos finally is in the grips of the demonic spirit, and he turns to Orpheus, growling like some sort of beast.