When an employee of a US enterprise switch their status from US lawful permanent resident to US citizen, must they report the change to their US employer?
No. There is no practical difference for an employer between a permanent resident and a citizen.
Employers do have to check employees' authorization to work, but they do not have to maintain on file a record of the current basis of each employee's authorization. They need to reverify authorization for employees whose authorization expires, but permanent residents, like citizens, don't have to worry about that. For permanent residents and citizens alike, the I-9 form is simply a record of the document shown to the employer at the time the employer checked, which is the beginning of the employee's employment.