Connection of functional derivative with variational derivative: $\frac{\delta}{\delta\phi(x)} F[\phi] = \frac{\delta F[\phi]}{\delta\phi}(x)$. Note that the variational derivative carries an extra coordinate variable dependence. It helps to make it explicit when there is similar confusion.
Functional derivative Leibniz rule: $\frac{\delta}{\delta\phi(x)} F[\phi] G[\phi] = \frac{\delta F[\phi]}{\delta\phi}(x) G[\phi] + F[\phi] \frac{\delta G[\phi]}{\delta\phi}(x)$.
Special case: $F_x[\phi] = \phi(x)$, $G_{i,y}[\phi] = (\partial_i\phi)(y)$, and $$\frac{\delta}{\delta\phi(z)} F_x[\phi] G_{i,y}[\phi] = \delta(x-z) (\partial_i\phi)(y) - \phi(x) \frac{d}{dz_i}\delta(y-z)$$.
Notice the distributional coefficients in the derivatives. There is no way to get away from them if you wish to consider $\phi(x)$ and such as functionals in their own right.
If you are interested in the BV formalism in the physics formalism, where the distinction between the functional and variational derivatives is barely remarked, I recommend the reviews by Henneaux and by Gomis, París and Samuel: doi:10.1016/0920-5632(90)90647-D, doi:10.1016/0370-1573(94)00112-G. If you are interested in the BV formalism purely from the point of view of jets, without bringing functionals into the picture, other than peripherally, I recommend this sequence of papers by Barnich, Brandt and Henneaux: arXiv:hep-th/9405109, arXiv:hep-th/9405194, arXiv:hep-th/0002245. If you are more interested in the BV formalism more from the functional point of view, with the appropriate level of functional analysis included, and with jets appearing only peripherally, I recommend the papers by Fredenhagen and Rejzner, as well as Rejzner's thesis: arXiv:1101.5112, arXiv:1110.5232, arXiv:1111.5130.