Encoding set of At-Most-One constraints as a MAX-SAT problem
The standard way to create soft constraints in MaxSAT is to use label variables:

For each $AMO_j$ constraint, create a new variable $l_j$. Then create an unit clause $(\lnot l_j)$ with weight $1$ and add the literal $l_j$ to every clause of the standard $AMO_j$ encoding that contains only hard (infinite weight) clauses. 

Now the label variable $l_j$ acts as a switch: setting $l_j = true$ will "turn off" the $AMO_j$ hard constraint, but does not satisfy the unit clause $(\lnot l_j)$.