M.H.
I went to florida with 2 infant twins and a toddler. I can't speak about southwest, but the rules are usually consistent across airlines....
FIRST-
You are supposed to inform the airline in advance that you are bringing a baby on your lap with you, or even an infant in a paid-separate seat. You need to call the airline and have them note your reservation. this is thet ime to ask if the plane looks full... if not, you may ask at the gate about moving to a seat that is next to an empty seat. ( but with such a large group, you probably should just stay together)
SECOND
Infant carseats and strollers do not count as a carry-on and should go with you all the way to the gate. A diaper bag also does not count as your carry-on. (1 per baby) You really should take your infant carseat if you can. The rental is something like $30 a day for a rental car's carseat... and your baby still should sit reverse facing. You need to make sure the rental carseat can provide reverse-facing for your baby's size...
NEXT
Request early boarding, don't wait for someone to make an announcement. (Some airlines did away with any official early boarding because it was abused, but if you ask for it , they usually wave you into the plan early without an announcement.)
You should be able to push your stroller down the ramp of the entrance and unload right at the airplane door .... this way you can get settled before everyone is piling into the plane. they need to tag your stroller and carseat for storing in the plane. This takes place right at the gate.
Note that it is sort-of rude to request early boarding for your entire party... Just you and your husband can handle the baby and get settled....
I would have a few different snack options with you for the baby, and two or three bottles of formula/breastmilk..... there is no limit to the liquid formula supplies for the baby. But it has to be prepared .. you can't take water separately. I found my babies drank more than I expected during the trip. If you have any flight delays, you'd want to have some extra to tie you over.
Also ,bring two outfits for your daughter in the carry-on....yes two... and few diapers and a whole bag of wipes... and diaper creme.
you just don't know when poop is going to leak out of a diaper amidst all the excitement. OR, some spill you never expected... or both. Also, a plastic bag or two, to hold any sudden stinky items that need to be contained.
Then try to feed the baby right around takeoff.. this helps the ears pop and maybe they fall asleep for a while from a full belly and all the humming noise while in flight.
I've had it both ways ... The screaming baby for 3 hours .. .and also the sleeps perfect riding companion. People are pretty nice about it, IF they are a parent too. good luck.