How are Latin botanical names handled in Chinese?
It seems they use both. On this site, alternativehealing.com, there is a long list of botanical names, the characters and the pinyin as well. For completeness, I must say that in the Chinese wikipedia for "botanical names", 學名 (植物), it's stated that Latin must be used for classes, see the part titled "學名來源".

On the same site, in this page, there is even a dictionary; you can look up by the Latin name, the pinyin or the number of strokes of the characters (if you don't know the pinyin but you know the character).