Including the operation $A\mapsto A^T$ can be viewed, in the language of operads, in many different closely related ways: via adjoining a new unary operation $J$ that satisfies $J^2=id$$J^2=\operatorname{id}$ and $J(ab)=J(b)J(a)$; via considering operads over the semisimple algebra $\mathbb{C}[t]/(t^2-1)$, via splitting everything into symmetric/antisymmetric, and considering the corresponding coloured operads (this looks like what you are doing in the examples of identities you give) etc. For either approach, you will find some papers dealing with similar things, though not necessarily literally the structure you are asking about. One way to try and list the possible identities would be to use operadic Groebner bases, - this way, for example, it is possible to show that for pre-Lie algebras the symmetrised operations does not satisfy any identities (http://arxiv.org/abs/0907.4958), but if there are identities, then one can detect them too, using an appropriate ordering. (It's like Groebner bases in the case of commutative algebras: if you are solving a system of polynomial equations and suspect that on all solutions one of coordinates $z_i$ has finitely many values, Groebner bases can detect that, and produce an equation in one variable that $z_i$ satisfies.)
