IMO you would be barking NOT to buy a seat, the bassinet positions arent guaranteed, and are likely too small for a 1 year old, who anyway will be mobile or at least squirming around and unlikely to stay put in a cot for 10+ hours. Even a child seat would be not great but at least you dont have a dead weight on you for hours.
Anyway I'd also say you should give it 3 or 4 years. A 1 year old wont get anything out of Florida (nor will any child under 3 ) , also not sure when you'd go but it can be blisteringly hot so then you are stuck keeping him or her out of the sun all the time.
If its a trip to see The Mouse you wont be both be able to go on each ride, so you will either go on half of them for the 4 year old or one of you will miss out a lot.
If its a Disney trip why not go when both children can benefit, if its not, there are plenty of closer by places you can get sunshine and sand and sea without such a long and expensive journey.