I was shooting my M&P340 at the range yesterday. It is basically the same gun as your 340PD but with a steel cylinder instead of Titanium. Mine weighs 2 ounces more than yours at 13.3 oz. as stated by S&W. I was shooting standard 38 special rounds with 158 gr FMJ bullets. I would shoot a cylinder and then switch to my L frame or N frame. After the pain subsided I would shoot another cylinder out of the 340. I shot about 40 rounds total that day out of my 340. My thoughts are for a fun range session, I don't want to deal with the pain. Even though I shot about 20 rounds of 357 mags through my L frame to finish up a box, I couldn't bring myself to put even a single round through my 340. The pain of the standard 38 specials wasn't overwhelming but I just don't see any positive benefit from pushing it. I felt that if I was ever in a self defense situation, I would be able to handle the pain and it probably would not be noticeable until after the situation had been dealt with. I didn't have any +P rounds with me but I think I could be effective with those from my 340. I have shot 357 mags from my steel framed J's and truthfully I don't wish to do more than a cylinderful of those.
I have the minimal Crimson Trace 405 grips on my 340 (it came that way) and think that the Uncle Mike boot grips might feel a bit better to me so next time I will try those. Incidentally, I had the laser switched off while practicing, preferring to use the Iron Sights.
In summation, for practice, minimize the pain. Whether you can shoot 1, 2, 3 or 5 shots at a time, then stop, take a break and shoot something more comfortable before you come back to your 340PD. I would recommend getting a steel framed J, like a 640, to switch off to. But I think practicing with any revolver will provide more benefit than switching off to a Glock. My range recently got some portable moveable stands for targets so yesterday, I was shooting my J at 10 feet (the closest I have ever shot it) and then switching off to my larger revolvers at 25 yards. For the first time, I tried shooting my J from the hip and missed the target entirely on my first shot, then I adjusted and was hitting center of mass. It was an interesting learning experience. (To anyone reading this, be sure of your backstop.)