Can I use the local x,y transforms of a plane?

You shouldn't bother with converting the points into 2d coordinates. Just using some vector math you can directly calculate the circle center point. In the process you also get the normal of the plane the circle is located in. I quickly created this method:

 public static Vector3 CircleCenter(Vector3 aP0, Vector3 aP1, Vector3 aP2, out Vector3 normal)
 {
     // two circle chords
     var v1 = aP1 - aP0;
     var v2 = aP2 - aP0;

     normal = Vector3.Cross(v1,v2);
     if (normal.sqrMagnitude < 0.00001f)
         return Vector3.one * float.NaN;
     normal.Normalize();

     // perpendicular of both chords
     var p1 = Vector3.Cross(v1, normal).normalized;
     var p2 = Vector3.Cross(v2, normal).normalized;
     // distance between the chord midpoints
     var r = (v1 - v2) * 0.5f;
     // center angle between the two perpendiculars
     var c = Vector3.Angle(p1, p2);
     // angle between first perpendicular and chord midpoint vector
     var a = Vector3.Angle(r, p1);
     // law of sine to calculate length of p2
     var d = r.magnitude * Mathf.Sin(a * Mathf.Deg2Rad) / Mathf.Sin(c*Mathf.Deg2Rad);
     if (Vector3.Dot(v1, aP2-aP1)>0)
         return aP0 + v2 * 0.5f - p2 * d;
     return aP0 + v2 * 0.5f + p2 * d;
 }

It returns the 3d center position of the circle as well as the plane normal the circle is located in. I simply used it like this inside Update: p0, p1, p2 and C are Transform references to 4 sphere objects.

     // inside update
     Vector3 norm;
     var center = CircleCenter(p0.position, p1.position, p2.position, out norm);
     if (!float.IsNaN(center.x))
     {
         C.position = center;
         var r = p0.position - center;
         int count = 100;
         var q = Quaternion.AngleAxis(360f/count, norm);
         for(int i = 0; i < count; i++)
         {
             var p = r;
             r = q * r;
             Debug.DrawLine(center+p,center+r, Color.yellow);
         }
     }

The result looks like this: