The expression to play one's hand is idiomatic and very common. There is no need to substitute it with another unless its author wanted to appeal to a wider and more international audience, some of whom might be unfamiliar with its meaning.
Astonishingly, I did not find a dictionary reference that supported the game of cards inspired phrase. The closest I found was in Merriam-Webster's entry for hand
9 a (1) : a player in a card game or board game (2) : the cards or pieces held by a player • studied her hand
b : a single round in a game • lost the first hand but won the next two
c : the force or solidity of one's position (as in negotiations) • trying to strengthen their hand
As a suitable alternative, the adjective shrewd, and its derivatives would be appropriate here. It carries a positive connotation and suggests that the person or persons are cautious and wary of making any mistakes, but ultimately they are thinking about gaining the best advantage for themselves, which I think is a common perception many westerners have of the Chinese government.
While this was happening, Beijing shrewdly kept a low profile
With some tweaking, one could use any of the phrases suggested below
- displayed shrewd leadership
- [their low profile] was a shrewd political move
- a shrewd player
Cambridge Dictionary defines
shrewd, shrewdly, shrewdness: having or based on a clear understanding and good judgment of a situation, resulting in an advantage: