Kudos, Emma, for considering all aspects of your options. In some ways, having attended an ivy can be a big advantage, but admittance to graduate school isn't necessarily one of them. Grad schools want well-rounded classes just as undergraduate programs do and they aren't going to fill their slots with all ivy-leaguers, which means that being one of few applicants from a non-ivy honors programs could even set you apart. The graduate program is going to consider the strength of the program you completed--and there are plenty of great non-ivy options out there--but your personal stats will be more important, i.e. your MCAT scores, GPA, research, co-curriculars, leadership, etc.
Keep in mind, too, that completing a medical degree can be very expensive. If merit scholarships help you keep college costs to a minimum during your undergraduate degree, then you are more likely to complete your degree and will be less burdened by debt when you begin your career.