I do not believe there is a tool or function that allows you to export a table with reordered fields. If anyone knows better please enlighten me! I've always felt it has been a bit of a limitation that the geo-processing tools do not honour the reordered fields. Especially the conversion tools .
In the past I had attempted to create a tool that did it. The following Python code can be used to re-order fields in a table by creating a new table with the fields re-ordered. In this example output (the new table) is sent to an IN_MEMORY workspace.
inputTable = "ABC" # This is a table loaded into ArcMap
outputTable = "XYZ"
# Get field mappings of Input Table
fieldMappings = arcpy.FieldMappings()
# Create an empty FieldMappings Object
newFieldMappings = arcpy.FieldMappings()
# Add fields in desired order. Note field index must be known
# Create Table
arcpy.TableToTable_conversion(inputTable, "IN_MEMORY", outputTable, None, newFieldMappings)
Dan Patterson recently created an interesting blog page on geonet showing how numpy can be used to do clever things with a FeatureClass. This could be another way one could alter the field order.
But ideally ESRI should make their tools aware of field order...