I blamed a restaurant for the booking I forgot. How do I correct the group chat?

The host asked whose name the reservation was under, and I gave mine even though I already knew what had happened. I'd opened the booking page that afternoon, got distracted, and never finished.

Six of us were standing behind me for my friend's birthday. I said, 'They must have lost it.' We ended up eating somewhere else, and the birthday friend spent the walk reassuring me that it wasn't my fault. I let her.

Last night she put a message in our group chat saying she was going to complain to the restaurant because they'd spoiled the start of the evening. Someone else said they'd avoid the place. Nobody has posted a review or sent a complaint yet, as far as the chat shows.

I typed 'Please don't, it was my mistake' and then deleted it. Admitting I forgot would have been embarrassing for about thirty seconds at the door. Now I have to admit I watched everyone blame people who hadn't done anything.

Do I correct it in the group where I made the claim, or tell the birthday friend first? I want to stop the complaint today without turning her birthday into a performance about how guilty I feel.