It is possible. Select positive signs for all factors but the first. The product of all these factors is then the generating function for partitions without 1's. Since this sequence is sub-doubling, G Pasemans' answer to question
https://mathoverflow.net/questions/248422/will-this-greedy-algorithm-always-work
shows that we can select the signs in the first factor so as to make the coefficient $a(n)$ be $1$, $-1$ or $0$.