It sounds to me like you are experiencing what an SG does. They're not
bad guitars, but the design places the bridge in the wrong position for there to be any profound bass response.
Part of what gives guitars their bass "punch" is the mass underneath the bridge, and part of what provides that mass is not just the body-wood and thickness, but how close the bridge is to the "waist" or the "hips" of the guitar body. Because it is a short-scale guitar and has the neck joined to the body so high up, the bridge on SGs has always sat either at the narrowest point on the body, or very close to it. The Les Paul, in contrast, has always had the bridge closer to the hips. And Telecasters, even with their skinny pickups, still get plenty of bass by situating the bridge almost at the hips.
This is not a "flaw" in the SG design, by any stretch. Indeed, plenty of drive pedals and even amps make a point of aiming for a distinctive tone by shaving off bass in strategic ways. My point is that you should be realistic in how much you think a pickup change will alter bass "punch".