My guess is weight is immaterial to how long you can hold a shotgun on point in an HD situation, which is not likely to last long. However, that extra weight will reduce felt recoil.
You mentioned the 870 Synthetic 7 shot, which is what I have. If I were concerned with weight, I would go with the basic 5 shot with an 18.5" barrel.
Felt recoil should not be that mucy of a factor, I would think, since you are going with a 20 gauge and it will only be used for home defense. If it is used for home defense, I think you will firing a few boxes of ammunition to familarize yourself with the gun, as opposed to an avid range shooter of which some blow though hundreds of shells in a day or week. That is where felt recoil is more of a factor, I believe, The avid shooters can explain much better.
Rent or borrow an 870 with a 12" stock before making up your mind.