I’ve shot a good number of .357 loads with my 14.9 oz. S&W 360. I don’t think it’s near as brutal as everyone says. Definitely packs more punch than the .38s, but I’m not dying to stop after a 5 round cylinder. And I’m not lying or not telling the truth.
^ I'm not as skilled of a wheelgunner as I'd like to be.
I'm not particularly recoil-sensitive, so a box-full of full-house .357 Maggie isn't something that I'd shy away from, particularly if someone handed me a box of it and said "go to town!" 
But the truth of the matter is that with a timer and shooting for score, objectively, I can't deal with (because of my inexperience) that much recoil - not effectively. As a concession towards that, what's loaded in my M&P 340 (which, IIRC, is only an ounce lighter than your 360) are Speer's "Short Barrel" flavor 135 gr. GDHPs in .38 Special +P.I really believe that part of what makes the .357 Magnum so formidable is actually "the Indian" sending that "arrow." The stats of single-round duty/defensive stops with the .357 Magnum are impressive to say the least, but I really do think that may well reflect the type (read: level of skill) of shooter who specifically choose to carry/load this caliber.
One of my favorite autopistols to-date remains a Coonan .357 Maggie that I rented at The Range at Austin, back in May of 2017 -
I was really lucky they allowed me to rent and shoot, there. The only reason they vetted me past their "no single shooter rentals" rule was because I had a valid Ohio-CHL, and one of their floor clerks whom I struck up a causal conversation with (as I was on the way out, actually, from initially having been told that they didn't rent to single shooters...without having asked about CHL-status) had also trained at one of my favorite local facilities, and we shared some overlap in friend groups.
Anyway, that's the long way 'round of saying that I was *very* pleasantly surprised by the Coonan. I honestly don't think that it shot any more harshly than any steel-frame, full-size .45 ACP single-stacker...which honestly is to say that it really was rather pleasurable to shoot. I wish I took a picture of my target, but that thing was clover-leaf'ing at 7 yards, easy.
I've also shot .357 Magnum in full-size revolvers in my youth, and I also found those experiences very enjoyable, with no particular recall of harsh recoil. Sadly, with my usual daily attire, I really can't achieve the depth-of-concealment I'd like, with a full-size revolver.
Other obligatory TX photos for y'all's enjoyment - Franklin's BBQ. Totally worth the 5 hours I stood in line for. And yes, I ate 4 pounds of meat in one sitting that day......
