The 3-2-1 method can be used in a large displacement nonlinear solution for a shell element model. The problem for solid elements would be the stress singularity that is created by selecting three points. But in a case with reaction forces that are very close to zero, that would not be a problem.
An eigenvalue analysis will run with a 3-2-1 method of constraint.
There are many reasons a nonlinear solution fails to converge. You have to look at why it failed to converge. What is the specific error message near the end of the Solution Output text listing?