I have no idea what you just wrote. If just want a Universal Health Care system vs a Single Payer, then just say it. Make all providers, doctors, nurses, hospitals, anybody remotely involved in health care, government employees or government health centers.
If you still want a Single Payer System, then you can do a few things to reduce fraud.
1. Make the system more consumer friendly. Get the consumers educated.
2. Change Medicare from a pay first chase later system, to investigate claims first, then pay.
3. Employ better processes and better technology. If private insurance companies have lower fraud rates, fucking buy there software and systems.
4. Make the punishment for fraud so horrific, doctors, nurses, lawyers, and roving ethnic bands you hear about will think twice about committing fraud.