Both of those options will help you get into the aerospace engineering field. If you're staying on the mechanical engineering track, make sure to also take electives related to aerospace as well as get internships in that field.
If your ONLY reason for transferring is solely because your school doesn't have aerospace engineering, I would say it's not worth transferring. If there's other reasons like the new school is more prestigious, is more well-known, is in a location that you like more, has more rigorous coursework, has more internship opportunities, or any other reasons, then I would say sure, go for it. In the end, both of these options are relatively even as it'll take you about the same amount of work to achieve both.