A vaginal infection that returns right after finishing an antibiotic your culture showed as sensitive usually means the infection wasn't fully cleared, or there's a source keeping it going, such as a partner who also needs treatment, or bacteria forming a protective biofilm that antibiotics struggle to penetrate. Burning and lower back pain returning this soon is worth a fresh look rather than just stopping here.
Next Steps
A repeat vaginal swab culture will show if the same bacteria persists or if resistance has changed. Doxycycline, which your earlier report also showed sensitivity to, is one option your gynaecologist can consider, alongside checking whether your partner needs treatment too.
Health Tips
See your gynaecologist soon if the burning spreads, you develop fever, or pelvic pain increases, since an infection that keeps recurring needs a more thorough workup.