these days the Bio side is flooded and competitive. Likely due to PhD's leaving the bench and looking for other careers.
You are speaking about a specific facet of IP law, namely patent law. More specifically, it seems that you're addressing a particular position within of the practice of patent law, which is that of a federal patent officer. If you are planning on going becoming a patent lawyer, your undergrad background will likely count very much. You might be able to use your undergraduate background to leverage a position at a better firm coming from a lesser law school. The days of few IP lawyers are over. I estimate that by the time you and I are practicing, IP lawyers will almost be a dime a dozen. They most certainly will be in 15 years.
What I think may give you an edge is studying EU IP law, or comparative Asian IP law. IP is intrinsically borderless, so the intersection of international jurisdictions and treaties and IP will be a hot area over the next five years.
Sorry if I'm taking this topic on a bit of a detour, but how is the demand for Industrial Engineers in Patent Law? As an 'imaginary' engineer with plans on going to law school, I'm interested in patent law, but I don't know how I'd fair with my background.