Reading the OP and ignoring all the responses, I'd be inclined to have you consider something based on the .223Rem or the .308Win. I will assume that an AR/MSR platform is out, so I will assume you are looking at bolt guns.
Frankly, for FL deer, the .223 Rem is not a bad choice. With the proper bullets and loading, you can most assuredly shoot to 1000 yards with it. There are factory configured rifles with stable stocks and faster twists to stabilize the heavies. There is always the .22-250, which pumps up the volume, but with an associated reduction in barrel life.
The .308 does have more recoil, but it is manageable. My 85 pound 12 year old does fine with it. The .243, .260 or 7mm-08 would be good choices based on your stated uses if the .308 is not appealing. The caveat is finding a factory offering with a sufficiently fast twist to stabilize the 105s in .243, 140s in .260 and 160s in 7mm-08. They are out there, but not prevalent. When you look at .308, almost everyone makes a fast twist .308 in a decent stock.
IMHO, THE budget build do-all hunting/precision medium range target rifle is the Remington 700 AAC-SD (.308). By medium range, I mean from about 400 to 1000. This rifle has a crappy $60 stock that needs to be replaced, but it will still hold 1.5 MOA in factory trim. The barrel is faster twist, it has a removable box mag, has a decent adjustable trigger, and it has a threaded barrel. You can add a comp or a suppressor. Adding a decent $200 stock, you will be at about $800 to $900 and cut group sizes in half at least. Add a decent scope (I prefer the Burris lines) and you actually have a rig that is good for all lower 48 game, can shoot 1MOA (or better) out to 1000 and you can cut recoil with a comp or suppressor. Stock weight is about 7 pounds, with an optic and a good aftermarket stock figure on 9 pounds.
Stock fit is important, and with a good aftermarket stock, you can get it fit for you and eliminate some issues. If you are not comfortable behind your rifle and can not repeat the NPA hold, you won't shoot with precision.