That said, prophylactic antibiotic therapy is not unreasonable.
Augmentin has not been studied in this contact, but
for sure it would successfully abort incubating syphilis. It's activity against gonorrhea is less certain, but 1,000 mg should effectively prevent or abort gonorrhea if taken twice daily for 5 days. The other STD of potential concern is herpes due to HSV-1, although the chance of that is very low.