S&W should change out barrels for you, but they will also put the entire gun back into stock, standard form when they do...if your trigger work is not within factory spec, best do not send it to them. Any decent revolversmith can do it, but finding a barrel may be the most difficult part. S&W will not...to the best of my knowledge...sell magnum barrels or cylinders although they will install them for you. (The reasoning behind this is to prevent some schmuck from converting his pre-war .38 Special into a .357 magnum) Yes, you can sometimes...rarely... find new S&W magnum barrels and/or cylinders for sale, but they are not a glut on the market. CDNN had some 686 cylinders a while back as I recall.
And the 629 is a .44 magnum, so if, indeed, you have a 629 and not a 686 you will be severly challenged to find a barrel for it since there are not the numbers of these guns out there that exist with the "K" and "L" frame .357's. Of course, another option is to have the barrel shortened, but that is a whole 'nother story and will require replacement of the entire front sight assembly. I have had three revolver barrels shortened in my life and only one job was satisfactory to me, but maybe I am just unlucky.
Me? I would sell or trade this gun and get what I wanted...the trigger is not as big an issue as the bucks you will spend for a barrel swap, and you will come out cheaper. JMO