It sounds like you charged the customer in error.

The surcharge is TG's way of charging its customers. UA does not charge their customers. If TG wants more money for award seats, they need to raise the price they charge to partners. UA will pay that price.

UA would then have to either eat it, charge a fuel surcharge, or raise a fee. It's up to UA, but it's irrelevant to TG, TG gets their money.

We have thousands of instances every day of carriers charging, not charging, or charging different fuel surcharges. All are dictated by the ticketing carrier (and kept by them as gleff points out).

For example, VS charges a fuel surcharge, CO does not on VS tickets. DL charges on their own, but CO does not for DL. US charges for UA tickets, but UA does not for their own or US. Then if you want to get into the details about the fuel surcharges, different carriers charge different amounts depending on the terms of their programs. That's because it's a ticketing carrier fee. On the back end, the partner carrier is being paid for the ticket, whatever the partner carrier's agreed upon price is.

In none of these cases is someone charged an add-collect while checking in.

In this case, the flyer held confirmed space with a valid reservation ticketed by a partner, and it appears the local station wanted to try playing audit police without having the background in the nuances of how these sorts of things worked. He had a valid reservation, with a matching ticket, the checkin agent's job is to make sure those two match, print the boarding pass, affix his bag tags and wish him a pleasant flight.

Honestly, if your job were to audit these sorts of things, you would have been trained in it and known the answer and wouldn't have had to ask how it works.

You probably should try to track down the customer and process a refund, as it appears he was overcharged by TG at checkin. And then use your customer service job to maybe give him something for the trouble. After all, customer service, not ticketing accounting, is your job.