I think this does only work for so-called homological vector fields, i.e. vector fields of degree 1 which self-commute. Then you have $[v,v]=0$, which is the Maurer-Cartan equation in $T_{poly}$, and which insures that the corresponding derivative $d$ squares to zero. 

Dealing with a Maurer-Cartan element you can simply use it to twist Kontsevich's formality $L_\infty$-quasi-isomorphism. When the Maurer-Cartan element is a vector field we know the explicit form of the first Taylor component of the twisted $L_\infty$-morphism: it is not HKR, but involve Bernoulli numbers (see e.g. https://www.math.ethz.ch/u/calaqued/research/LecturesDufloETH.pdf). 

Anyway, the $k$-th taylor component of the $v$-twisted $L_\infty$-morphism will be given by the series
$$
\phi_v^{(k)}(u_1,\dots,u_k):=\sum_{l\geq0}\frac{t^{l+1}}{(l+1)!}\phi^{(k+l)}(u_1,\dots,u_k,\underbrace{v,\dots,v}_{l~times})
$$
To conclude one just have to observe that $\phi^{(k)}$ preserves the grading given by the arity minus $2$ (arity means the number of arguments for poly-vectors and poly-differential operators). Therefore each time $v$ appears in the formula it decreases this degree by $1$.