This issue is no longer reproducible in ArcGIS Pro 2.6. The FieldMappings parameter must be an input parameter, and set the dependency on a table or feature layer/class parameter. Load the FieldMappings parameter value with an arcpy FieldMappings object in UpdateParameters validation:
def updateParameters(self):
if self.params[0].value:
fieldmappings = arcpy.FieldMappings() fieldmappings.addTable(self.params[0].value)
self.params[2].value = fieldmappings
The Field Map parameter will load and functions properly in the geoprocessing tool dialog box.