Recommend the .44 Magnum...BUT!....
BE prepared for a surprise! I have Contender barrels in .22LR, .22 Hornet, .222Rem, 9mm, .357mag, .44 Mag, .45 Colt, .45 Win Mag (newest one, haven't shot it yet), also .30-30 and .45-70.
For hunting there are two main things to balance. Portability (size/weight) vs. recoil. T/C barrels come in a huge variety of configurations. Most common are 10" and 14". The 10" barrels are found in round and octagon. Round barrels are heavy. Octagon barrels are very light. My 10" octagon .44 Mag (iron sights) has more recoil, and a lot more muzzle jump than a 6" .44 Mag revolver! It flat amazed me! Eventually I figured out that the 10" octagon T/C is considerably lighter than a .44 mag revolver, so it made sense.
Also, the grip of the T/C allows for a different feel to the recoil than that of a revolver. There is no second shot to worry about maintaining your shooting grip for. I prefer the rubber Pachmyr grips over the stock wood, but to each his own.
Add a scope or red dot sight and the weight goes up, so the felt recoil comes down a bit. Add a longer (14") or heavier (round) barrel and get more recoil reduction, but at the cost of some portability. A scoped 14" contender is not a small package, and a hip holster isn't practical any more. There are some good shoulder/chest holsters for these guns, allowing you to use both hands while moving around.
The main advantage to choosing the .44 Magnum over the other listed cartridges is power, cost and commonality. The bigger, more powerful rounds are more expensive (and .44mag is plently expensive enough), and less likely to be found on a dealer's shelf. You can get .44 Mag at Walmart (if hey have any in stock) odds are you won't find .460 S&W anywhere but a well stocked sporting goods/ gun shop, and maybe not even then. The answer to this is to simply order the ammo, online, or from the shop. .44 Mag brass and bullets are also common where ever reloading components are sold. Again, not cheap, but less than the specialty cartridges.
While I am recommending the .44 Mag, you might consider if the .357 Mag will fit your needs. With the proper load, the .357 has enough energy for deer out to 100yds, provided you can deliver a single precise shot. A scoped Contender will do that, if you are skilled and practiced with it. Recoil is a lot less than the .44, and ammo and components are cheaper as well. BUT, you must chose the correct load for deer (and place your shot right, as well for a humane kill), the 125gr .357 bullets are intended for humans, not deer. 158gr (hard cast or jacketed) or heavier is the way to go for deer. Some folks speak highly of the 180gr .357 bullets, but I have no personal experience with them.
The biggest real advange to the Contender (the old ones, not the G2) is a good trigger pull, but many are very stiff to open. The G2 is reported to not as good a trigger (but still quite usable) and they are easy to open. Also the G2 allows you to recock the hammer without opening the action, something the original Contender does not. If you scope the gun, get a hammer extension. It will come in handy.
Besides being one of the most accurate handguns available, the ability to get another barrel for a lot less than the cost of another gun (in my area usually less than $300 and sometimes less than $200 used) is a huge plus. You could get a .357 and if you find it isn't enough, get the .44 barrel. OR vice versa, if the .44 is more than you want/need, get the .357. Keep or sell the spare barrel after you have tried them out. Aother plus is, if using a scope, the scope mounts on the barrel, so your zero remains pretty constant (as long as you don't bang the barrel/scope around) when you switch back and forth between barrels.