My LO is 3 weeks old today. She had tongue AND lip ties. We got hers lasered when she was a week and a half. They swaddled her, put numbing cream on the areas and it was done in less than ten minutes. I wasn't in the room once she was swaddled bc I wasn't allowed but she barely made a peep. And the moment they finished with the tongue she was sticking it out and moving it all around bc she never could before. I put her to the breast within five minutes of it being done and we've been fine ever since. She nurses normally now, we never even needed pain meds for her. And she's finally gaining weight, slowly, but gaining! And she wasn't before. She's only 3 weeks but it made all the difference for us.
Edit to add: is there a reason they are doing anesthesia for your LO? It seems pretty overkill. I had lip tie and had it lasered around age 14 and it was so quick and relatively painless. I still remember it. Just curious.