I'm no expert on operads but it seems that

* a "bracket system" can be formalized as an [operad](https://en.wikipedia.org/wiki/Operad); see e.g. [the Poisson operad here](https://mathoverflow.net/a/261448/61197),

* the "product-complete" condition could be related to having a Hopf operad (see previous link),

* the "Lie-complete" condition says derivations are inner (I don't know how to phrase this using operads).

You may also be interested in the notion of [algebra over an operad](https://ncatlab.org/nlab/show/algebra+over+an+operad), more tangentially the notion of "convolution Lie algebra" associated to an operad, and operads defined by using graphs (as in graph complexes).

Note that in the Poisson case, a derivation being inner means that it is a Hamiltonian vector field. You claim that every derivation defined by a bracket expression (with numerical coefficients?) is inner. I understand the example with the Jacobi identity, but why should it be true in general? A necessary condition for a vector field to be Hamiltonian is to vanish on Casimirs of the Poisson structure; this is clearly satisfied. But consider for example $f\{g,h\}$. It is a derivation with respect to $h$, but it is impossible to solve $f\{g,h\} = \{Q,h\}$ for $Q$ in general (using only the axioms) if $f$ is not a Casimir (if $f$ is a Casmir then $Q = fg$ works). (For example on $\mathbb{R}^2$ with $\{x,y\} = x$ it is impossible to solve $y\{x,-\} = \{Q,-\}$.) The space of Hamiltonian vector fields is a module over the space of Casimirs, not over the space of all functions (in general).