During RESIDENCY? BME is a very difficult 4-year degree. Given the pre-med requirements you have certainly completed, that could be cut to two years. But to try to do that simultaneously with a full-time residency would be next to impossible.
Unless you reach PhD status in engineering, the degree won’t get you that much more status. I would urge you to read up on the topics that attract you, metallurgy, material science, etc, and become better versed in those than the average doc, assuming you have adequate free time to do so. You might even be able to get some streaming courses from the major universities to aid in your quest.
FWIW, I was a BioEngineering major in college…mostly EE, but I took a MatSci course as well. That background has served me well in Radiology and NM, particularly with the advent of PACS.