IME, the pleasant perception of the very light weight of the 340PD is quite noticeable for CCW as compared to, say, the S&W Airweights or Ruger LCRs.
As one who at one point had the unrealistically-optimistic perspective to shoot quite a lot of premium commercial .357 Magnum through these little guns (I have two of them), I'm sure that almost any variety of typical .38 Special WC (except perhaps Buffalo Bore's 150gr WCs) should seem reasonably non-punishing for any reasonable amount of practice.
"Carried much, shot little" is a good way to regard these little pistolas, and if your chosen WC load in fact hits to POA, the extra felt recoil due to the lack of gun mass is certainly worth it, at least to me. (I carry mine with .38 Special Gold Dot short barrel 135gr+P JHP, since that load hits to POA.)
OTOH, if you were required to practice frequently and extensively with heavy .38 Special+P loads, I'd certainly pick a heavier "snubby", regardless of the extra discomfort for CCW.