I am SO glad you got to a vet; cat scratches can and do kill snakes because of bacterial infection. The meloxicam and chlorhexidine sound fine - I assume the meloxicam is injectable and that the vet showed you how to do this?
I'm very twitchy about that oral children's antibiotic. It's a broad-spectrum antibiotic that is given to snakes, but not in that form. I don't like to undermine trust in your vet, but injectable antibiotics tend to be the best delivery system for snakes, and I frankly have never heard of giving a children's oral antibiotic to a snake (I worked as a vet tech, but it's been a long time).
In your shoes, I would ask your veterinarian to set up a telephone consult with a veterinarian associated with the Association of Reptile and Amphibian Veterinarians. You can use the link Bogertophis provided to find one in your county or state.
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