I am consider a window unit for the baby's room. I worry about bugs getting in though.
Nothing to worry about. As long as you put it in the window correctly no more a chance of bugs getting in than from a screen. Depending on how much you guys like the heat (and where you are in FL) you could just use window fans as well. My mom lives in FL but LOVES the heat. So they don't turn on their A/C at all until July and turn it off by September at the latest. (Of course she is a wuss on the other side and turns on the heat as soon as the temp drops below 70.)
But look at the costs overall for whatever solution you go with. It may be better to just use the central system and be a little more badass than your neighbors the rest of the time (so set it to 80 or something and use as little clothes on your kid as possible, etc). Full disclosure I am not a parent. And I am not suggesting putting your child in danger. But kids are tougher than some parents/adults give them credit for and 80 shouldn't be an issue assuming they are otherwise healthy and hydrated.