@iulianc, that is not advice I would personally give.
Although changing majors within the CENG is certainly possible and many have done it, there is critical information that we are missing when looking at those anecdotal examples, mainly the MCA score of those who were able to complete that task.
One of the criteria that they use (see the whole policy below) is this: “a. Eligibility for the proposed major at the time of admission.” That means if they didn’t have the MCA to get into a specific major as a first time freshman, they may be denied. CP does this specifically to avoid back dooring into a more difficult admit via the route you’ve suggested. Swimming down stream (e.g. from ME to IE) is much more common and easy that swimming up stream (e.g. IE to EE). Moving from IE to EE, although possible, could depend on the original application. It is not without significant risk.
“http://www.catalog.calpoly.edu/academicstandardsandpolicies/otherinformation/#ChangeofMajor”