How about raising the garage door just far enough for ventilation purposes, but less than anyone could crawl under?
Most garage doors will allow hot air to travel out through the top when raised a little bit.
With the alarm on, that should prevent unwanted visitors in the garage.
As a Floridian myself, I can testify that raising the garage door just those few inches really helps with the heat. One of those small high-intensity fans blowing out under the crack bumps the effectiveness up noticeably, too.
Lock the door between the house and the garage, and add whatever alarms and/or cameras are in your budget. You are trying to prevent opportunistic thievery - someone who is willing to snatch something from an open garage and get away quick, but not willing to work or risk much to steal. The noise from an alarm is likely to be a decent deterrent.