Just for the fun of it: I have never heard of EE is being the backbone of design, etc. But, I maybe wrong. Every time in any conversation talking about designing things, they always refer to ME not EE.
Just for the fun of it, regarding the designing thing, I will copy the definition of ME and EE from wikipidia:
Mechanical engineering is the discipline that applies the principles of engineering, physics, and materials science for the design, analysis, manufacturing, and maintenance of mechanical systems. It is the branch of engineering that involves the design, production, and operation of machinery. It is one of the oldest and broadest of the engineering disciplines.
The key word: design is in this definition.
Electrical engineering is a field of engineering that generally deals with the study and application of electricity, electronics, and electromagnetism. This field first became an identifiable occupation in the later half of the 19th century after commercialization of the electric telegraph, the telephone, and electric power distribution and use. Subsequently, broadcasting and recording media made electronics part of daily life. The invention of the transistor, and later the integrated circuit, brought down the cost of electronics to the point they can be used in almost any household object.
Electrical engineering has now subdivided into a wide range of subfields including electronics, digital computers, power engineering, telecommunications, control systems, radio-frequency engineering, signal processing, instrumentation, and microelectronics.
The key word: design is not found in this definition
Don’t get me wrong: Of course, you can force EE major to do design too if you will. But, it is not common.
So, just for the fun it, if OP wants to do BME with lots of design then ME is the answer meanwhile other majors like EE, biology, chem or chemE, or bio material science (related to Tissue Engineering in BME), they also can do other BME concentration.
So, OP we have given you lots and lots of information in here. Do you still want to go to BME for grad. school or you prefer monkey around with the systems (hardware and software) in Comp. Science major?