The problem is unclear. I think you have a labelled graph G=(V, E) with labels c:V->R, such that each node in V consists of a triple (L, R, S) where L is a sequence of weights are on the left, R is a sequence of weights that are on the right, and S is a set of weight that have been taken off. Define c(L, R, S) to be the centre of mass. Introduce an undirected edge e={(L, R, S), (L', R', S')} between (L, R, S) and (L', R', S') either if (i) (L', R', S') results from taking the first weight off L and adding it to S, or (ii) (L', R', S') results from taking the first weight off R and adding it to S, or (iii) (L', R', S') results from taking a weight from W and adding it to L, or (iv) (L', R', S') results from taking a weight from W and adding it to R.

There is a starting node (L_0, R_0, {}) and an ending node ({}, {}, W) , with the latter having L=R={}.

I think you're trying to find the path (L_n, R_n, S_n) from the starting node to the ending node that minimises the maximum absolute value of c(L_n, R_n, S_n).

I won't post a solution, as requested.

You are overthinking it.