I wanted to post an update in case any other mommies are looking for the same thing.
My husband and I decided on a UK Family Medicine (located at 740 S Limestone) doctor for our daughter instead of a pediatrician. Dr. Keisa Bennett and her team are open to slowing down the vaccination schedule.
It's not always easy to get an appointment with Dr. Bennett (you may have to see a member of her team) but she has a great team and it's worth it. Plus, my daughter is receiving great care and I'm not given a hard time for my choices. She is up to date on her shots, only receives 2-3 shots at a time, and has consequently not had any adverse reactions to the vaccines.
I highly recommend any UK HMO mommies in the same situation consider checking out UK Family Medicine.