I would think a .357 magnum would have better stopping power than a 9mm especially at point blank
If there is any significant difference, it's not going to be that much in either direction.
Today's premium hollowpoint rounds are designed to penetrate and expand according to specific specifications. You're probably going to find that a 9 mm premium hollowpoint round will have about the same efficacy as a 357 Sig premium hollowpoint, a .40 premium hollowpoint, etc. The question is if the extra recoil of a more powerful caliber is going to be worth a very small increase?
although nowhere near the stopping power of a .30 06 rifle round.
Again, apples and oranges. For that matter, it's probably closer to comparing apples to turnips.
So I would go with a .357 magnum hollow point. Preferably a critical defense round by Hornady.
On what grounds? What platform?
The .44 magnum is a bit too hard to control.
This is not necessarily true at all. If you use a .44 magnum in the way it was intended to be used (longer barrel, nice solid N-frame platform), then the recoil from a full fledged .44 magnum load isn't going to be difficult to control at all. Too many people buy into the Dirty Harry hype...
Now, while the handgun might be a poor manstopper unlike a good .30 06 rifle round or a 12 gauge round, I would think a .357 Magnum would stop just about anybody from close range provided you hit them, and by close range I mean around 5 feet or closer not 20 feet away.
Again, you're taking some poor guesses here. I think you'll find with a bit of reading and testing, that just about any commonly used handgun round (9 mm or greater) used in a defensive setting is going to do about the same job, regardless of whether you are 5 feet away or 20 feet away (they don't lose nearly as much velocity from that distance as you may believe).