Gerstenhaber’s deformation theory might possibly give cohomological obstructions for extending say an A_3 structure to an A_4 or maybe even A_infty structure.
There are some remarks on formality and the de Rham complex on p 29, which may possibly be relevant. Can we frame our problem as one of formality for the Deligne complex??
Keller p 4: If A is and A-infty algebra and f: A –> V is a qis of complexes, then V admits a structure of A-infty algebra s.t. f extends to an A-infty qis.
nLab page on Operad notes