https://pubmed.ncbi.nlm.nih.gov/37210378/ found beta-lactamase in a small percentage of bartonella (<2%). I suspect this is not representative of bartonella spread by pet cats that have been given antibiotics.
My experience with bartonella was that many antibiotics had minimal or no effect, but I got a severe herx with certain antibiotics (especially atovaquone/minocycline).
I never tried methylene blue though, and headaches weren't my primary symptom, so I don't know if that's a herx or side effects of methylene blue. I had side effects from cryptolepis, so I didn't take that long enough to find out if it did anything to bartonella.