I'd have to give my vote to the J frame. I've had a 638 for a little over a year now and carry it in a OWB fobus. You'll forget you have it on you even after all day carry.
I find the j frames hard to shoot accurately, but not sure its an issue at BUG ranges, I had a 638 and sold it and of all the guns Ive had and sold its the one i regret selling the most
Get the Smith. I like the Sigs but, for a BUG you want something thats for sure, no safety you won't have to worry about, jamming, etc, etc. The Smith has a lot of power for a primary gun let alone a BUG. Don't undercut yourself on this one.
I have both a S&W 38 and a Sig 238. I carrythe Smith in a Don Hume pocket holster and it carrys very well. The sig, on the outher hand, is carried in a Don Hume belt slide. I wouldnt want to chose between them as I like them both.