A few questions:

Do you have any hunting experience with a handgun? How about pistol shooting in general? Also, what are you intending to hunt with your .44?

Remember that .44 Magnum in a revolver is nowhere near as powerful as a typical deer rifle of the .30-06 class, so shot placement is the key {of course it is with rifles, too!!}. Even critters shot well with a handgun will often run and require tracking. As Gustavo says, don't expect a handgun to be a stopper. Treat it like you would a bow and wait for the perfect shot. And heavy sixguns take quite a bit of shooting practice to develop skill guaranteeing adequate shot placement.

I found the AE 240 to be a very accurate factory load in my Ruger Blackhawk. I chronographed it at 1350 fps.

The trouble is that with HP's or JSP's you may experience poor expansion. Several years ago I had a bad experience with 240 Noslers HP's {not Partition} for just that reason. That bullet failed to expand at all in one bear I shot with it. Performed like a FMJ. Subsequent testing in my bullet media produced bullets looking identical to the one I was able to recover from the bear, and showed almost no deformation of the bullet.

I got better results with the Remington 240 in a handload at 1100 fps so there is a difference. Note that the strike speed of 1100 is not a full-house load. I never tested any at 1300+ in my media as I was shooting it in a very light 6.5 inch barrelled S&W Heritage revolver, so there is always the chance that the extra 200 fps would cause the bullet to come apart but for deer I wouldn't worry about that too much.

At typical .44 Mag velocities it is pretty hard to argue with the recommendation for a Keith type bullet in the 240-300 grain range. Such bullets offer an edge in tissue disruption that you may not get with round-meplat jacketed bullets and the Keith bullets typically also provide deep penetration.

It is reasonably easy to test your prospective jacketed bullets to determine if you will get any expansion in game. Many jacketed bullets don't expand reliably and due to their somewhat rounded configurations may not give you the terminal effect you want and you may be better off with a big flat-nosed Keith bullet.