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While I would agree that PI is an appropriate charge and probably let it drop, the officers had a good prima facie case for criminal trespass. One of the legal requirements is crossing a fence line that is designed to keep people out or livestock in. Since he entered the backyard through a gate in the fence, this legal requirement is met.

I think the attitude of the officers sounds abominable. I know exactly how busy it gets on a New Year's Eve. One officer could have stopped for a minute and explained to the OP why they would charge for PI and what the requirements for trespass are. If the OP wanted to file the charge, the officer could have done it during the booking for him. This is simply good customer service and treating the victim of the crime politely.