Totally understand what you mean regarding school holidays.
If I was choosing between the dates you say it is an easy answer, October is (in my opinion) one of the best times to go so that would be my choice.
To be honest if you are able to I would avoid July and August, whilst the summer holidays are the obvious time to go it's extremely hot this time of the year, in addition July can be very crowded too.
There is never a bad time to go to Florida just some months better than others, ideally a combination of nice weather but not too hot and manageable crowds is what I would look for if I was able to choose when we go, we are stuck with Easter or July/August - so next year we go in March.