Changing the order of fields in an existing geodatabase feature class or table has been challenging since the early days of ArcGIS - ArcMap and ArcCatalog. Without bothering to dig up all of the historical references, blog posts, issue reports, bug reports, great idea postings, workarounds, previous methods and tools, etc, I point at one simple solution to this 16 year old problem (it's old enough to drive in most US states).
Purchase and install Sparx Enterprise Architect (EA) version 12.x or higher, Professional edition or higher, floating or single use license.
if you spend a few hours that your employer bought from you trying and failing to re-order geodatabase fields, your employer has very well paid for the EA software and has nothing to show for it. << justification for your request.