Disclaimer: the following is my practical experience on research. I have no idea about the opinions of admissions committees and the like, but it probably depends greatly on the country and the relative quality of master's and undergrad education.
I am a theoretical physicist by formation (5 years degree). Then I did my master's thesis (I had taken enough courses) in Bioinformatics, and I am doing a PhD in Biophysics.
When I start a new project (or tackle the next sub-project), there are a bunch of things about Biology and Chemistry that I don't know, but most of them are actually quite easy. I am sometimes lacking "the bigger picture", being able to my questions into a broader meaning, but that is not so important. For example, my master's project was to improve the number of identified peptides in an experiment using computational techniques, and that is a very clear goal. What to do with this improved results? Obviously, we can improve the things we already do. But, are there biological questions it can help answer? I am not sure, but there are experts on that.
For the middle picture range, I rely on my advisors. They are also physicists, but they have learned along the way pretty much what they need. And after just a few months, I was surprised of how much I was able to help the new members of the lab.
Actually, I believe taking another undergrad degree would not help so much. Of course, I would be faster at the beginning, because I would know what things like "the $C_\alpha$ residuals" mean, but the actual meat of the project, where we spend months, is probably not covered (or not covered in enough detail) in most undergrad degrees. And the main reason is that these details are known only to those who have actually worked hands on with it.
Let me give you an example: in Physics we talk about spectra all the time, and all the information you can extract from it, with perhaps, the only limitation is the noise of your instrument. The truth is, unless you have a VERY expensive camera, you are going to find very funny stuff, like two spectra taken right after the other, with exactly the same experimental set up, will not have, on camera, the same intensity; and even the profile of the line. To compensate for this you need to get clever, and it very much depends on the details, so it is very difficult to teach unless your lecturer is an expert in spectra analysis and can tell you how they do it. And still, most people can just rely on the spectra pre-processed by the experts, so they don't need to know this. Unless, of course, you want to work with raw spectra yourself (been there).
Lastly, a master's or a PhD has some courses. They are usually quite specialised, targeted for your level and background, and can bring you up to speed in the things you need to know about your field quite nicely.
And to add some peace of mind, my former lab hired a postdoc coming from computer vision. His biological knowledge had quite big holes, but nevertheless, in a couple of weeks he was already doing amazing stuff with very good ideas.
Bottomline, go for the advanced studies. You can always take Biology or Economy on the side (for example Open University or unofficially at Coursera).