This seems to bolster the theory that autism is modulated by the gut-brain axis. Being given antibiotics at such an early age will probably severely dysregulation gut microbiota. And GABA actually does cross the BBB but in small amounts. Perhaps at such an early age, dysregulation in GABA produced in the gut has significant effects on brain health - over stimulation, learning memory issues, etc.
It’s extremely common for young kids to receive antibiotics. In some countries, antibiotics are over the counter and many parents will give their kids antibiotics for nearly any infection. Antibiotic misuse is rampant in some countries where they aren’t gated behind prescriptions.
Any such link with autism would therefore be an extremely rare side effect. The rate of antibiotic use in children is far higher than the rate of autism.
I don’t think this case supports the antibiotic theory by itself at all. I think it’s confirmation bias because antibiotics are one of the current trending theories among mainstream discussion.