I dig what you are talking about Hdiddy. My bass does sound a little bit less puffy with the Underwood in the wings. I guess in my case/my bass it doesn't bother me so much, very slight difference at the most. I did a CD for a jazz singer in Hong Kong a while back, no pickup, just a great mic, a Sanken from Japan aimed at my bridge, great studio with a very capable staff who were very into getting a good bass sound. It turned into several sessions, on different days. Most of the time I removed my Underwood, but on one of the days I decided not to bother, just left it in, unplugged. Once I heard the finished product the bass sounds the same on each tune. I can't hear the difference and can't remember what tunes I had my pickup installed and which ones I didn't!
By the way, I recently posted a very favorable report about hearing Eddie Gomez playing a borrowed bass, not a fine carved bass but a Shen, probably plywood, a hybrid a best. He made it sound great with a BP-100. Enough so that I'd like to try one on mine. Not that it would make me sound anywhere near as good as he sounds!