The Tamil name "தெற்கு பச்சுளி" (terku pacchuli) sounds okay. "தெற்கு" (terku) usually refers to the direction South. Whereas, the adjective 'Southern' can translate to "தென்" (then), "தென்னக" (thennaga), "தெற்கத்திய" (therkathiya) or "தென்னிந்திய" (thennindiya), or sometimes தெற்கு (terku).
So, in this context, "தெற்கத்தி பச்சுளி" (therkathi pacchuli) or "தென் பச்சுளி" (then pacchuli) would be more fitting, but if you prefer "தெற்கு பச்சுளி" (terku pacchuli) that would be fine too.
The spelling patchouli can be written in Tamil as பச்சௌலி.
Best,
Vijayasankar