That's a very expensive option. You might save money by just doing full day private charters and having the captain run you back and forth a few times.
If you use google, there's a plethora of rental operations.
If you're not an experienced boater, do research first.
There are free classes a plenty. I'd be apprehensive about driving a boat out in the keys myself. I've been on board a professionally captained boat in KL that has run aground (Rainbow Reef). I've also run aground myself here in my home town. Driving a boat is more complicated than it looks.
One other thing.
Don't assume the rental companies will give you everything you need to be safe. The last time I rented a boat, I found that the fuel gauge didn't work and neither did the darned radio. The speedometer worked intermittently. You do actually need to know your speed, to deal with slow speed "zones" and such on your way in/out. Luckily I didn't have any fuel problems and didn't actually need the radio since I stayed within cell range of land that day as planned. I wouldn't do that in the ocean, no way. Find out what the safety items are and verify they all work yourself. Don't skip over seemingly obvious things like guages - although it's really hard to tell if they work until you get underway in some cases.
Get yourself a nautical gps map app. Download the map files for it in advance. This can help with a broken speedometer, too!
Here's how to perform a radio check. Sea Tow services the keys, so you can use their service to check the radio for free:
Automated VHF Radio Check | Sea Tow
Hidden Costs can be huge: When you rent, find out how much they charge for fuel and how much the boat uses. The first time I rented a boat I was blown away at how much fuel it used. Just couldn't believe it. I think I paid around $400 for a day's rental and another
$450 for gas! There's a few other fees they usually tack on (insurance and whatnot, you DO want the insurance.. did I mention it's easy to run aground?)
Here's the biggest hurdle in my mind:
Unless you and your father are taking turns solo diving, you'll be anchoring with nobody to watch the boat. One big wave and it could be gone. It happens. Also, it's easy to get into currents in the keys. Without anyone on the boat there's nobody to come get you or call for help if you "lose" the boat. Luckily it's busy enough that you stand a chance of getting aid from another boat. Do you have a third person who can bubble watch while you dive? That's immeasurably safer.