It seems that you were looking for a blend of two stories in the same collection: "Poor Planet" by J. T. McIntosh and "Shamar's War" by Kris Neville. Both can be found one after another in the collection Seven Trips Through Time and Space, edited by Groff Conklin.
In "Poor Planet," the planet's name is Solitaire. The main character is a spy sent to find out why it has remained a poor, low-population and low-tech planet for as long as it has, when by theory it should have prospered. He does have special buttons, one of which communicates with his ship. He meets a girl in a music shop (looking for an old version of an opera), and they are both captured by the local security force - but escape, due to buttons. The planet is cold but the park is in a heated dome, and there is a significant scene there.
It is immediately followed by "Shamar's War." That story features a planet named Itra with the natives called Itraians, in which the main character is also an undercover offworlder (sent to foment a revolution). It seems like a reasonable mistake of memory to blend them a bit if you read them one after the other.