Personally you are making this way harder than it is.
If you let them know within time frame as required by your job (like 2 hrs before shift starts sounds reasonable) or as stated in your employee handbook
FMLA Family Medical Leave Act
Continuing treatment by a health care provider that results in an incapacity (inability to attend school or participate in other daily activities) of more than three consecutive calendar days with either two or more in-person visits to the health care provider within 30 days of the date of incapacity OR one in-person visit to the health care provider with a regimen of continuing treatment, such as prescription medication, physical therapy, etc. In either situation, the first visit to the health care provider must occur within seven days of the first date of incapacity. Examples include pneumonia, surgery or broken/fractured bones.
Employees may take FMLA leave for themselves or to care for their parent, spouse, son or daughter whose medical condition meets the above criteria. The FMLA regulations specifically exclude the following conditions, unless inpatient care or complications develop that would meet the above criteria: cosmetic treatments, common colds, flu, ear aches, upset stomach, minor ulcers, headaches other than migraine, routine dental or orthodontia problems, and periodontal disease.
Of course if you have track record of abusing time off, well you might have to have other documentation.
Most health providers are more than happy to oblige you with providing documentation if the need arises.