I have a bachelor's in applied mathematics. I would like to know is it worth it to go straight into a graduate degree for electrical engineering or should I get a bachelor's first?

I would like to go into signal/image processing, but also want a graduate degree in electrical engineering. Should I first get another bachelor's degree in electrical engineering and then go on to graduate school for it, or should I go straight into the graduate program for it. I have read that that plenty of people go into engineering graduate programs without the undergrad but am curious about your opinion on it. My grades and GRE scores are good enough to get into Columbia, and the admit officer there told me that they love applied mathematicians since they know they can handle the mathematics at the graduate level. They told me they find regular engineering majors struggle with mathematics at the graduate level.

My biggest fear I would venture to say is getting a job. Would it look bad with an undergrad in applied mathematics and a masters/ PhD in electrical engineering or should I finish the bachelors first?

Thank you in advance.