I'll preface this post by saying that I'm not suggesting that anyone do what I have done, I am only telling you about my experiences. What any of you do is strictly up to you.
Now, a waaaayyy back in 2008 when I bought my 360J specifically to use it for a 9mm conversion project I called S&W to get some info just be sure that I wasn't having a grenade built. At that time I was told that the 360J frame is the same one that is used in the M&P 360 .357 magnum. I also specifically asked about the cylinders and if there was any difference in the metallurgy or heat treating of the .38 special cylinders compared to the .357 magnum cylinders and was told that the only difference between them was the depth that the chambers were cut.
Given that information and knowing that SAAMI max pressure for the .357mag is 35k PSI and having read numerous accounts of successful 9mm (35K PSI max) conversions that had been done using 940 cyls in early 642s, as well as knowing of several 940s that had been reamed to 9x23 to use .38 Super (36.5K PSI max), and having owned a PC940 356TSW (48K PSI in the same cyl as the 940 except for chamber dimensions), and subsequent info I got from the gunsmith, I went forward with my conversion plans.
When speaking to Mark Hartshorne of Pinnacle High Performance prior to having him do the work on my gun, he pointed out that, unlike the solid 4"-6" test barrels, snubbies not only bleed off pressure from the barrel/cylinder gap but that their shorter 1 7/8"-2" barrels don't allow for full pressure build up (which is why bullets are slower out of shorter barrels).
So, now that all of that background info is out of the way, I will tell you that I have shot 9mm, 9mm +P, 9mm +P+, .38 Super, 9mm Largo, and 9x23 Winchester through my 360J over the last 8 years with no ill effects. In fact, I had the gun back to Mark last year to have a titanium .38spl cylinder fitted to it and he went through it thoroughly and the gun is in fine shape. Now, in the interest of full disclosure, it's been far more regular pressure 9mm than the others, say 85%, followed by 9mm +P+ and .38 Super 10%, then 9mm +P and 9mm Largo 5%, and just a couple of cylinders of 9x23 Winchester because the recoil is incredibly brutal out of a 13oz gun.
All of that being said, I personally would not try to load .38spl cases to .357mag pressures, you're just inviting catastrophic case failure. Because, while the gun may be built to handle .357mag pressures, .38spl cases are not made to withstand them. If I wanted to shoot .357mags out of my 360J I would either get a .357mag cylinder for my gun or have my existing cylinder reamed to .357mag dimensions and use .357mag ammo/cases.
Finally, before anyone gets all up in arms about changing calibers, cylinders, etc., you need to keep in mind that S&W and other gun companies standardize many components in the interest of saving money. It's cheaper in the long run to have one frame built to use in many guns that is built to the strength necessary for the most powerful chambering compared to building multiple versions of exactly the same frame that are tested to different strengths - weaker frames for weaker cartridges. The same goes for cylinders, less expensive to have all of your cylinder blanks made and heat treated to a certain strength then cut the chambers as needed than make a bunch of stronger or weaker cyls. It may have been different 60, 70, 80+ years ago but this is how it is now.
In addition, there is a long history of rechambering revolvers or converting them by changing cylinders and barrels. In the '60s, '70s, and into the '80s, N frame revolvers in some calibers, like .44mag,, were extremely hard to come by. It was quite common for gunsmiths to convert guns like the M28 Highway Patrolman, which were plentiful and inexpensive, to more desireable calibers by changing out cyls and barrels or rechambering and reboring them. It was a matter of making do with what you had vs what was available and the desired outcome vs cost effectiveness. Obviously, this practice continues today with gunsmiths like Mark Hartshorne, Hamilton Bowen, David Clements, and others offering interesting and unique caliber conversions. In fact, just the other day I shipped an M28-2 to Mark along with a titanium .41mag cylinder and a 4" M58 barrel, when he returns it to me I'll have a slick and useful midweight 3" round butt .41mag revolver in place of the beater .357mag he's starting with.