The only way I know to construct a formality quasi-morphism for poy-vector fields out of an homotopy is via Tamarkins approach (i.e. $G_\infty$-formality). What Tamrakin does is

1. prove that there is a suitable $G_\infty$-structure on Hochschild cochains

• prove that the obstruction to construct a $G_\infty$-formality step by step is unobstructed. At each step you have to make some choice, and the homotopy for the Hochschild complex of De Wilde-Lecomte gives you a way to do such choices.

But Tamarkin construction (part 1) involves the choice of an associator (i.e. choice of appropriate weights in Kontsevich's $L_\infty$-qausi-isomorphism)... so I think that this is hopeless to construct the formality out of an homotopy.

Moreover, even the other way I don't see how you could associate an homotopy for the Hochschild complex to an $L_\infty$-quasi-isomorphism from $T_{poly}$ to $D_{poly}$.

I might be wrong but I have the feeling that you are mixing two different notions of homotopy: that of higher homotopies in the $L_\infty$-morphism, and that of homotopy retract for the Hochschild cochain complex. In particular, what do you mean by a "homotopy equivalence" between $L_\infty$-algebras (whatever you mean, a homotopy for the Hochschild complex in the sens of De Wilde-Lecomte won't produce an example).

The only way I know to construct a formality quasi-morphism for poy-vector fields out of an homotopy is via Tamarkins approach (i.e. $G_\infty$-formality). What Tamrakin does is

1. prove that there is a suitable $G_\infty$-structure on Hochschild cochains

2. prove that the obstruction to construct a $G_\infty$-formality step by step is unobstructed. At each step you have to make some choice, and the homotopy for the Hochschild complex of De Wilde-Lecomte gives you a way to do such choices.

But Tamarkin construction (part 1) involves the choice of an associator (i.e. choice of appropriate weights in Kontsevich's $L_\infty$-qausi-isomorphism)... so I think that this is hopeless to construct the formality out of an homotopy.

Moreover, even the other way I don't see how you could associate an homotopy for the Hochschild complex to an $L_\infty$-quasi-isomorphism from $T_{poly}$ to $D_{poly}$.

I might be wrong but I have the feeling that you are mixing two different notions of homotopy: that of higher homotopies in the $L_\infty$-morphism, and that of homotopy retract for the Hochschild cochain complex. In particular, what do you mean by a "homotopy equivalence" between $L_\infty$-algebras (whatever you mean, a homotopy for the Hochschild complex in the sens of De Wilde-Lecomte won't produce an example).