What are the ethical implications of lying to get into a course?
If I found out I was intentionally deceived by a student after telling them they could not take my class, I would alert my dean (I would go beyond the department chair) that said student was violating the rules and spirit of the exchange program. At a minimum I would expect that student to no longer be able to take advantage of the exchange and have their grade withheld from the other class. I would hope the home university would also punish this student for academic misconduct.