If you really want to do a lot of swimming type of stuff, then pick September.
April will be more crowded, and it may be too cool to enjoy swimming. I have been in Spring twice and both times we swam once but it was a cool swim. But, I went in March, so you might have better luck a couple of weeks later in April. We have a pool at home but my kids really wanted to swim in the Disney pool.
I felt like it was hot in September and I don't really ever want to go back in that heat. Especially not with a hot,sweaty baby strapped to me. We still had a fun time, but I'm not a fan of the heat. March was very nice weather-wise (minus the swimming). It still rained off and on but we came prepared for that.
I've been with an 11 month old (was not yet walking on her own), a 4 month old, an 8 month old, and a 22 month old. The babies were a lot easier than the 22 month old. The 4 month old was difficult because she wasn't sleeping through the night very well yet, but 8 months and 11 months were a total breeze. Just bring an Ergo or other good quality ergonomic baby carrier. Bringing and cleaning bottles for babies was a pain in the neck, but when I took the nursing baby that was not a consideration. Disposable bottle liners also meant that I only had to wash nipples and not the whole bottle, too.
We still had a great time with the 23 month old, but I had planned everything out so we didn't really wait for much of anything. She could not have taken a lot of lines and even as it was she sobbed a few times waiting in the character lines over feeling rejected by her favorite characters (because she couldn't run to them immediately and had to wait her turn - it's hard to be that age and not understand anything).