First of all, you might want to look at the MO question How to define the equivalence of Maurer-Cartan elements in an $L_\infty$-algebra?, since of course this is effectively what you need (morphisms can be described as MC elements in a particular convolution $L_\infty$-algebra, I suspect you know that).
Let me also say that some sort of approach to homotopy of the kind you are asking for is discussed by Martin Markl in this paper, and in this paper it is checked that the sort of definition proposed by Martin can be obtained from a Sullivan-type approach to homotopies (outlined in an answer to your first question) using appropriate homotopy transfer theorem for homotopy cooperads.
It is worth remarking, in particular, that the definition of a "derivation homotopy" suggested in another answer to this question does not work: the crucial difference between $A_\infty$ and $L_\infty$ is that $A_\infty$ is a nonsymmetric operad, and therefore instead of the Sullivan's algebra of differential forms on the 1-simplex we can use the dg algebras of chains on the 1-simplex (which is noncommutative) - this gives the "derivation homotopy".
