Why wouldn't you do this like any other buckling analysis? Just apply a unit radial load to a free-free model and your first buckling mode should give the load necessary to elastically buckle the ring. Then check stresses to see if you are near yield in which case you will have to do non-linear run.
When modeling something like this for buckling, symmetry is not always your friend as it can sometimes hide some modes.
Just a thought, if you go to Smalley wave springs website
they have their wave springs modeled in Solidworks and you can download them.
This would at least give you a start to this model.