The BSD conjecture for an abelian variety $A$ over a function field holds if Ш$(A)[\ell^\infty]$ is finite for some prime $\ell$ ($\ell = p$ allowed). This is a theorem by Schneider, Bauer and Kato-Trihan. If $A$ is a constant abelian variety, Ш$(A)$ is finite by Milne's PhD thesis.
Edit: Since the analytic rank $\rho$ is always greater or equal than the algebraic rank, one has BSD if $\rho = 0$ (by the equivalence of weak BSD and the finiteness of an $\ell$-primary component of Sha). I show this inequality even for Abelian schemes over higher dimensional bases over finite fields in http://kellertimo.name/Height.pdf, Lemma 2.17.
