See Gesenius' entry 4 under Piel כִּחֵשׁ:
- to deny
- to lie
- to deceive (one's expectation)
- to feign, to flatter, most commonly used of the vanquished pretending subjection and love towards a victor
Someone who is feigning obedience would submit himself to the ruler, though only because he has no other choice. Adam Clarke compares this to Pharaoh's feigned submission to the God of Israel:
Thine enemies submit themselves - Literally, lie unto thee. This was remarkably the case with Pharaoh and the Egyptians. They promised again and again to let the people go, when the hand of the Lord was upon them: and they as frequently falsified their word.
Pharaoh was initially defiant at the command to let the Israelites go:
1 Afterward Moses and Aaron went in and told Pharaoh, “Thus says the Lord God of Israel: ‘Let My people go, that they may hold a feast to Me in the wilderness.’”
2 And Pharaoh said, “Who is the Lord, that I should obey His voice to let Israel go? I do not know the Lord, nor will I let Israel go.” -Exodus 5:1-2 (NKJV)
However after the second plague, he feigned obedience and lied by saying he would let them go:
Then Pharaoh called for Moses and Aaron, and said, “Entreat the Lord that He may take away the frogs from me and from my people; and I will let the people go, that they may sacrifice to the Lord.” -Exodus 8:8 (NKJV)
But as soon as Pharaoh saw that there was relief from the plague of frogs, he changed his mind and did not let the Israelites go (v15).
This happened several more times during the different plagues. The power of the one true God was so great that Pharaoh resorted to feigning obedience and lying in order to have the plagues removed (see Exodus 7-12 for the account of all ten plagues).
Feigning obedience is itself lying. And Pharaoh no doubt cringed, if only to himself, when he gave permission for the Israelites to leave, as any enemy would when he is forced to acknowledge a power that he can't overcome.
So the NIV is not actually incorrect here. Using "cringe" just portrays the initial stage of someone who just realized he now has to feign obedience to a ruler that he doesn't like and doesn't actually want to submit to.
Definition of cringe:
verb (used without object), cringed, cringing.
- to shrink, bend, or crouch, especially in fear or servility; cower.
- to fawn.