A flower is always a flower, a plant, and all the names of plant species are inanimate in Russian irrespective of how many anthropomorphic features the plant has, even in a fable, which means it will always have the Accusative case not equal to the Genitive case.
However, if Подсолнух is a proper name of a human, of a sentient plant with anthropomorphic features, or of any other animate being, then 1) it will be capitalized (Подсолнух), and 2) its Accusative case will be equal to the Genitive case.
Note, if you introduce подсолнух as a species different from the sunflower plant, e.g. if it will be, say, the name of an alien sentient species, then it will be used as animate, Accusative = Genitive. But in this case it will not be the same word as the usual Russian word подсолнух, the two words will be homonyms, and as the author you'll have to put effort into explaining the difference between the two species.