Sounds like it is a true recurrent pyoderma as opposed to a persistent pyoderma. That makes a tenacious bacterial infection less likely. So less need to focus on the antibiotic type or cultures and to look instead for underlying causality.
Again, time and allow the pet to get older or start on the aforementioned list of testing (food trial, blood tests, rescraping, dermatologist referral).
I hope that helps.