The best way to become a biomedical engineer without doing a full undergraduate degree. If your marks are adequate, you could apply to do a Master's in biomedical engineering. Check the programs where you would like to apply for the best fit for your undergraduate degree. For example, in the University of Ottawa in Canada, there is a Department of Electrical Engineering and Computer Science. The fact that these two fields are together makes it easier to fit a degree like yours.
I am sure some other universities have a similar combination of fields. After 2 years or so, you could be a qualified biomedical engineer.