In most situations, Eames isn't adverse to a tactical retreat. He's a much bigger fan of staying alive than he is having an uninjured ego, but in this case it's more than that. He has a responsibility, and moreover this is an insult he refuses to stand for.

"Surely you should be encouraging me to stay, make some other unwise decision," he raises an eyebrow, curious. "My death or my capture would be a decent win for your mistress."