I decided to not heed any warnings about replacement lenses for the GXV3672 and I looked around for a 2.8mm lens that was only 15mm deep. I found one on Ebay and got it in within a few days. The below pictures are before and after. You can see the field of view is much larger now, but I'm still having a focus problem on the new one. I doubt I'll be able to improve it much since I was concerned about tightening it down so far that I would crack the sensor. You can also see how the external case is vignetting the image. I could live with that if I could get it more in focus.
https://flic.kr/p/qKd8Hb
https://flic.kr/p/r2FQf5
So, short answer is, get the right lens to start. I'm going to see about getting a 3610 which comes with the 3.6mm lens. And also, I guess we need to clean the banister. Oops.