I take my little one to Dr. Mizrachy at West Broward Pediatrics. They have two locations, one in Weston and one in Plantation.
All of the doctors there are great but Dr. Mizrachy is especially pro-breastfeeding (I exclusively breastfed for the first year) and is also open to discussing vaccination schedules. They have a schedule they use but I remember him asking us at our initial meeting how we felt about vaccines and I don't see why he would have a problem with cloth diapers.
They are open late on Wednesdays and on Saturday mornings and will stay late other days to see sick children. You can call and set up a preliminary "interview" which is what we did and we fell in love with Dr. Mizrachy and have had nothing but good things to say about the other doctors as well. Their number is ###-###-####.