We took DS#2 when he was just 7 months. It was a wonderful trip, and he is high maintance--partially due to health issues. As far as the heat goes, we live just outside Houston. Hot, hot, hot and humid from March to November--we are currently experiencing an extereme cold front, it's 84. FL weather really isn't any different (and I speak from experience). So dealing with heat is just part of our everyday life. Here are my tips:
Dress in light clothing. Light material
and light colors. Don't layer--ie, skip the onsie and just put on the shortalls.
Wide brim hat. And light blanket. Not all cotton blankets are light, even those with holes. It needs to be breathable too. Keep that baby covered from the sun. I'll frequently have the baby in the front pack at that age with a LIGHT blanket covering them.
Some sunscreen should be OK. I remember when DS#1 was 5.5 months old, AAP came out and said babies under 6 months
could wear sunscreen. BUT, CHECK WITH YOUR PEDI. FIRST!
I have clip on fans for the stroller. I also have a sunshade; a big breathable thing that goes over the stroller like a rain cover that has holes in it (DS4 says it doesn't block the view) and keeps out the UV rays. It also manages to keep the stroller cooler. A big blanket while baby is sleeping would also work.
Lots of liquids. Nurse/bottle feed often. I actually introduced water at 4 months with DS#2, it was 113 out (plus 100% humidity) & he just didn't want to nurse, but downed the water.
Enjoy those inside attractions during the heat of the day if you don't go back to your room for naps/swimming. That's what Hall of Presidents was made for!

Really, Buzz, Carasoul of Progress, Festival of The Lion King, and many, many other rides work wonderful for this.
That's all I can think of for now. I hope that helps.