Practical Implementation for Refinement Order on Integer Partitions
I suggest you try using integer linear programming.  Introduce $k\ell$ zero-or-one variables $x_{i,j}$, with the constraints

$$\sum_j x_{i,j} \lambda_j = \mu_i$$

and

$$\sum_i x_{i,j} = 1.$$

Feed it to an off-the-shelf ILP solver.  If it can find a feasible solution for you, then you know that $\lambda \le \mu$.  If it can determine that no solution exists, then you know that $\lambda \not\le \mu$.