Learn English – An idiom for using power of position to get things done

You could say they would be playing into the hands of the enemy.

play into somebody's hands

to do something that gives someone else an advantage over you, although this was not your intention

If we allow terrorists to disrupt our lives to that extent we're just playing into their hands.

_{Cambridge Idioms Dictionary, 2nd ed.. (2006). Retrieved February 24 2016 from http://idioms.thefreedictionary.com/play+into+hands}

According to this source, the expression originated from playing cards, specifically the game bridge. The source gives nice outline of its emergence:

A part of the game’s strategy is to force your opponent to play certain cards. If you manage to do so, then she or he is playing into your hand, giving you an advantage. The expression is often pluralized into playing into someone’s hands. This slight modification has resulted in obscuring the origin of the meaning, making people think of a body part, when actually in card games your hand, refers to the cards you are holding. This meaning goes all the way back to the 16th century as well.

This idiom is very common in the UK, and is also often said with emphasis and pluralised as mentioned above:

played it right into my hands

Searching for the above on Google Books will give you plenty of examples of the use of this phrase in print.

https://www.google.com/search?tbm=bks&q=%22played+right+into+my+hands%22

I'm not exactly sure that the idiom of playing dead like a rat translates to being cunning, but playing dead/pretending to be asleep to avoid injury or confrontation has an equivalent in playing possum.

Your request sounds more like "playing [the] innocent":

"Forgive me, I do not understand," said Cicero, playing the innocent. - NYT
John is playing innocent, and he knows more than he is telling us.

Even more common is playing dumb:

To pretend that one has no or little knowledge (of something); to act ignorant or uninformed (about something).

Another is "dumb like a fox". Foxes are clever foes. When one is pretending not to be clever, this can be used. Also used, in a similar situation, is "crazy like a fox." But these imply hidden cleverness as a disguise, and @Edwin Ashworth's answer is a more common idiom for when really disguising yourself when you have predatory intent.