I'm 5'11" and around 300LBS (I'm trying to lose weight- I was 305 at my peak, last couple weigh-ins I was 295). While I do have a large-frame, unfortunately I do qualify as fat.
I find IWB to work just fine. When I'm not pocket carrying I usually carry IWB (often I carry in the pocket at the same time) when in a carry state. I like to carry IWB just behind my hip bone on the strong side. I find it comfortable, and it disappears quite nicely. I have spent the extra money for a decent gun belt for years now and it makes a big difference.
I have several guns, from 1911s to a Taurus PT140 Millennium Pro (Glock 27 sized) to J-frames that work very well IWB, hide very well, and are very comfortable.
I also like pocket carry. As big guys our pockets are often a little bigger than normal so this may work with a better variety of guns than for thinner guys. My PT140 almost works pocket carry- it fits, though the lump in the pocket is a bit thick and gun-shaped. Generally a lighter gun is more comfortable this way (doesn't feel like it is pulling down your pants) and I think it prints in your pocket less.
I really like my S&W 442 in a leather DeSantis pocket holster. Lightweight J-frames are nearly the perfect pocket carry gun. 15 oz, 5 rounds of .38+P, the lumps that may be shown by the gun are rounded and don't scream "gun" like the squared off corners of an auto, and they fit well. Occasionally I carry the Taurus 85CH that way, but at 21oz it prints a lot more than the 442.
Your intended use of your 637 in a pocket is a good choice. I'd also suggest buying a cheap Uncle Mike's IWB holster to see if you are OK with IWB carry. I do think IWB is the easiest to conceal (especially "tuckable" IWB holsters) and it is doable for us big guys. If you find the thicker guns like Glock or the Taurus Millennium Pro series to be too thick, there are plenty of thin CCW guns that should work well IWB even for a big guy. Check out 1911s, Kahr, Walther PPS, etc.