The first thing to clarify is that the Business Process Flow stage and the Opportunity status are not the same thing.
The BPF represents the guided sales process, while Opportunity status represents the record lifecycle, such as Open, Won and Lost.
For example:
If you need a custom Opportunity field to always reflect the active BPF stage, you can trigger automation when the BPF instance/stage changes and update the Opportunity accordingly.
For simple asynchronous requirements, Power Automate can work well. For strict transactional business logic, a Dataverse plugin is generally more appropriate.
Also remember that the standard Sales Pipeline view derives its pipeline phase from the active BPF stage, so I would first verify whether the issue is actually with the BPF or with a custom field/automation.