Final answer:
To accommodate Mrs. Smith while maintaining the schedule, it is best to inform her of the available time slots and assist her in choosing a mutually convenient appointment.
Step-by-step explanation:
The most professional and efficient way to handle Mrs. Smith's request for a vaccine appointment for her dog would be to inform Mrs. Smith of the available time slots and assist her in choosing one that is mutually convenient. This option respects both the client's needs and the constraints of the schedule. Mentioning the precise dates and times that are available for Fluffy's vaccine would help Mrs. Smith make an informed choice, and allows for the schedule to remain under control.