I have been carrying Speer 38+P 135 Short Barrel in my 442 Pro "Moon Clip", and the 638 before it. I also have 125 +P Golden Sabers as well as 125 +P Mag Techs. I am sure all 3 loads will work fine.
Also carrying a speed strip with 5 shotshells in my pocket for the smaller size snakes in the grass.
I also carry an LCR-357 and have it loaded with the 135 Short Barrel 357 Mag loading. It is not quite as brutal as the 340 to shoot with hotter loads due to the LCR's polemer frame flex as well as the Hogue Tamer Grip.
As I do not have an airweight 357 J Frame I have not been able to compair 357 recoil between the Ruger and S&W with me doing the shooting.
I have however shot them both side by side with the same 38 Special 125 +P Golden Saber load. The LCR does soaks up the recoil much better.
Bottom line on the 340 is I am not sure how severe the recoil would be with 135 Short Barrel 357 ammo. My understanding is that it is not as nasty as Midrange 125 grain 357 Magnum. More like a 38 Special +P+, or just a tad more.
Being an experianced shooter, my best guess is that you can probably shoot 5 rounds of it ok in a serious situation with no issue.
Now a down side. I do shoot my concealed carry gun quite a bit during the year. Hill Billy Concealed Carry Last week I shot 6 rounds of Stingers through my LCR-22 on a Copperhead. If you do shoot your CC gun during your everyday travels, it may be better to keep 38 Specials in it. I have no idea how loud a 135 Short Barrel 357 Magnum is out of a 2" barrel when behind the gun. I am guessing louder than one would care to shoot without carrying a set of earplugs in your pocket. Just thinking out loud.
On the other hand if you only shoot your CC gun on the range. I would probably practice with 38's and carry the short barrel 357's for personal defense.
Bob