Although my undergrad major was in Physics, I did almost all of my electives on electrical and computer engineering. Part-time work as a computer engineer was also available, which gave me valuable experience. This has not been a problem for me. The starting salaries for graduates of MIT in various disciplines were published that year. My physics salary was listed as the highest and it was higher than the average for EEs. While I believe my work experience was the most important factor, having a broad background was also crucial. While I was working, I did graduate work in EE while at the same time. However, most of my work related to lasers and radar.
Many of the technology leaders I met over the years had engineering-physics degrees. The majority of this work involves the development of new technologies. It was very helpful to have a background in physics.
If you are just looking to become a designer engineer, then the engineering degree will suffice. You can do both if you want to become more. Your career will be driven by your overall experience in school and after. The breadth of your experience will help you to build on that. The thinking process in Physics is different from engineering, and this can help.