I agree, instead of obtaining another degree you can focus on developing a small portfolio targeted towards ID, using some of your past work and current ideas. See if you can create some interesting sketches showing you can also handle the creative process to some extent. Then with your current degree and with a clear story of what you want and some good ideas and abilities regarding design, you could definitely land an ID job.
I know several people who have shifted from ME to ID and it works nicely. These people can be a very valuable addition for example in small studios where products have to be both conceptualized and prepared for manufacturing, the collaboration is very fruitful and they are not that separate of a discipline as you describe in my experience.