Unless it's possible to get a Papal Dispensation from the Laws of Physics, getting 'comfortable to shoot' & '11oz .357' in the same sentence is just not doable.
Be resigned that this is a gun to be carried a lot and shot a little. I have an older steel 640 that makes a 'practice' gun for the Scandium Magnum. You can put Pachymayr grips on until the revolver is nearly as big as a K-frame, and it is not going to totally solve the problem. 'Small' and 'light' is the plan with the 340; a plan that you'll defeat with huge grips.
A better solution, if you are determined to fire the little bugger a lot, is a half a pair of bicycle gloves, with one on your gun hand. The padded palm will draw a lot of the sting, and the exposed fingers let you operate the trigger. You don't wear shooting muffs on the street, and you won't wear one glove either, but give a bike glove a try to make your practice tolerable.
That's tolerable for 30-40 rounds. Unless you are a serious masochist, don't figure on all day at the range. All you'll get are sore hands and a huge flinch.