My son has faced the same dilemma. The schools he applied to that had an undergraduate biomedical engineering major were out of our financial reach, despite scholarships. He has decided to pursue a mechanical engineering degree at a college that offered him a full scholarship and admittance into an honors program. He hopes to either double major or minor in biology as well.
What I have done, though, is look at the curriculum of the schools he wanted to attend and hope to use those to guide him when it comes time to select electives for these majors/minors. This will hopefully help him gain admission into a biomedical engineering graduate program.
Another thing you might want to do is look at the types of research professors in the "regular" engineering majors are doing. For instance, at my son's college, there are two professors who do work in what might be considered "biomedical engineering", ie, prosthetics and rehabilitation products.
I have also looked at job openings at various BME companies to see what skills they require for job applicants. For instance, some require an applicant with knowledge in a specific computer language. Although that particular job is not applicable to my college entering son, it is a place to go to as you progress through college as to what computer classes you may want to take if they are not included in your curriculum.
As a parent, I was very pleased that my son chose to not take on a lot of debt in his undergraduate program, as he knew that he would definitely have to attend graduate school to get any type of job in biomedical engineering. I was also happy to not have to take on a lot of debt for him to go to a specific school for biomedical engineering, only to find he did not like that major, and had passed up full scholarships and wiped out our savings to spend one year to figure that out.
Good luck to you.